Advanced Quantum Mechanics, Lecture notes of Quantum Mechanics

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Advanced Quantum Mechanics

Judith McGovern

December 20, 2012

This is the web page for Advanced Quantum Mechanics (PHYS30201) for the session 2012/13. The notes are not intended to be fully self-contained, but summarise lecture material and give pointers to textbooks. These notes have been prepared with TEX4ht, and use MathML to render the equations. Try the link to “This course and other resources” to see if your browser is compatible. (In particular there is a problem with square roots of fractions that you should check is OK.) If you are using Internet Explorer, you may need to download “MathPlayer” from here. Please report errors to Judith McGovern.

This course and other resources.

The books I have relied on most in preparing this course are:

• Shankar, R. Principles of Quantum Mechanics 2nd ed. (Plenum 1994) [preview] [Errata to 13th printing, (2006)]
• Gasiorowicz, S. Quantum Physics 3rd ed. (Wiley 2003) [Supplementary material] [errata]
• Mandl, F. Quantum Mechanics (Wiley 1992)
• Townsend, J.S. A Modern Approach to Quantum Mechanics (McGraw-Hill 1992) [pre- view]

Throughout these notes, I have provided section references, mostly to the first three of these. Note that Shankar and Townsend use Gaussian rather than SI units, as do older editions of Gasiorowicz. Notes on translation are given in section A.12. I have found two sets of extensive on-line quantum mechanics notes that are at the right level for this course. One is by Prof Richard Fitzpatrick, of the University of Texas at Austin. His UG course notes are here and his PG notes here. The UG notes start at the level of our PHYS20101 course, but continue to the end of this course. The PG notes also start from the begining but more in the style of PHYS20602, and continue to the end of this course. They seem very much equivalent (though not quite idential) for the coverage of this course. The other is by Prof Jim Branson of the University of California, San Diego, notes avail- able here; these again start at the beginning, but go beyond this course and cover parts of PHYS30202 as well. Both sets of notes should be useful for revision of background material and as an alternative resource for this course. Both have more details in the derivations than I have given here. These notes have been prepared with TEX4ht, and use MathML to render the equations. If you are using Internet Explorer, you may need to download ”MathPlayer” from here. Once you have done so, the following should look more or less the same:

2 π ℏ |〈kf^ |eE^ ·^ r|k〉|

(^2) δ(E i −^ Ef )

∫ ∞

−∞

xne−αx 2 dx =

√

π α

The first is MathML, the second is an image. Subtle things to look out for are bold-face

for vectors k etc; not so subtle is whether the square root in the second equation covers both numerator and denominator. If you are using Firefox, you may need to download STIX fonts; see here. Safari (even 5.1) is still a mess, unfortunately.

The advantage of MathML is accessibility (at least in theory), equations that can be mag- nified if you change the magnification of the page (or that pop out with a click using “Math- Player”), and much cleaner directories. The disadvantage is that I don’t think it looks as nice on average, and I don’t know in advance how much trouble might be caused by incompatable browsers. I may revert to using images for equations (as the web notes discussed above do) if there are widespread problems, so let me know.

Operator methods in Quantum

Mechanics

1.1 Postulates of Quantum Mechanics

Summary: All of quantum mechanics follows from a small set of assumptions, which cannot themselves be derived.

There is no unique formulation or even number of postulates, but all formulations I’ve seen have the same basic content. This formulation follows Shankar most closely, though he puts III and IV together. Nothing significant should be read into my separating them (as many other authors do), it just seems easier to explore the consequences bit by bit. I: The state of a particle is given by a vector |ψ(t)〉 in a Hilbert space. The state is normalised: 〈ψ(t)|ψ(t)〉 = 1. This is as opposed to the classical case where the position and momentum can be specified at any given time. This is a pretty abstract statement, but more informally we can say that the wave function ψ(x, t) contains all possible information about the particle. How we extract that information is the subject of subsequent postulates. The really major consequence we get from this postulate is superposition, which is behind most quantum weirdness such as the two-slit experiment. II: There is a Hermitian operator corresponding to each observable property of the particle. Those corresponding to position ˆx and momentum ˆp satisfy [ˆxi, pˆj ] = iℏδij. Other examples of observable properties are energy and angular momentum. The choice of these operators may be guided by classical physics (eg ˆp · ˆp/ 2 m for kinetic energy and ˆx × ˆp for orbital angular momentum), but ultimately is verified by experiment (eg Pauli matrices for spin-^12 particles). The commutation relation for ˆx and ˆp is a formal expression of Heisenberg’s uncertainty principle. III: Measurement of the observable associated with the operator Ω will result in one of thê eigenvalues ωi of Ω. Immediately after the measurement the particle will be in the correspondinĝ eigenstate |ωi〉. This postulate ensures reproducibility of measurements. If the particle was not initially in the state |ωi〉 the result of the measurement was not predictable in advance, but for the result of a measurement to be meaningful the result of a subsequent measurement must be predictable. (“Immediately” reflects the fact that subsequent time evolution of the system will change the value of ω unless it is a constant of the motion.) IV: The probability of obtaining the result ωi in the above measurement (at time t 0 ) is |〈ωi|ψ(t 0 )〉|^2.

If a particle (or an ensemble of particles) is repeatedly prepared in the same initial state |ψ(t 0 )〉 and the measurement is performed, the result each time will in general be different

(assuming this state is not an eigenstate of Ω; if it is the result will be the correspondinĝ ωi each time). Only the distribution of results can be predicted. The postulate expressed this way has the same content as saying that the average value of ω is given by 〈ψ(t 0 )|Ω̂ |ψ(t 0 )〉. (Note the distinction between repeated measurements on freshly-prepared particles, and repeated measurements on the same particle which will give the same ωi each subsequent time.)

V: The time evolution of the state |ψ(t)〉 is given by iℏ (^) ddt |ψ(t)〉 = Ĥ |ψ(t)〉, where Ĥ is the operator corresponding to the classical Hamiltonian. In most cases the Hamiltonian is just the energy and is expressed as ˆp · pˆ/ 2 m + V (ˆx). (They differ in some cases though - see texts on classical mechanics such as Kibble and Berkshire.) In the presence of non-conservative forces such as magnetism the Hamiltonian is still equal to the energy, but its expression in terms of ˆp is more complicated. VI: The Hilbert space for a system of two or more particles is a product space. This is true whether the particles interact or not, ie if the states |φi〉 span the space for one particle, the states |φi〉 ⊗ |φj 〉 will span the space for two particles. If they do interact though, the eigenstates of the Hamiltonian will not just be simple products of that form, but will be linear superpositions of such states. References

• Shankar 4

1.2 Position and Momentum Representations

Summary: The position-space representation allows us to make contact with quantum mechanics expressed in terms of wavefunctions

• Working in one dimension for the moment, it is convenient to define the state |x 0 〉 which is an eigenfunction of the position operator: xˆ|x 0 〉 = x 0 |x 0 〉. Here we have used x 0 to indicate a specific value of x – say 3 m right of the origin - but we also write |x〉 when we want to leave the exact value open. It’s like the difference between |ω 1 〉 and |ωi〉.
• The set {|x〉} form a complete set as ˆx is Hermitian, but now when we expand another state in terms of these, we have to integrate rather than sum, and rather than a discrete set of coefficients {ai} we have a continuous set or function, |α〉 =

∫

α(x′)|x′〉dx′. (Integration limits are −∞ → ∞ unless otherwise specified.)

• We choose the normalisation to be 〈x|x′〉 = δ(x − x′). Then α(x) = 〈x|α〉. This is called the wave function in position space of the state |α〉. With this normalisation the identity operator is (^) ∫ ∞

−∞

|x′〉〈x′|dx′.

These states have infinite norm and so don’t really belong to our vector space, but we don’t need to worry about this! (See PHYS20602 notes for much more detail.)

• Since |α〉 =

∫

α(x′)|x′〉dx′, from the fourth postulate we might expect that P (x) ≡ |α(x)|^2 is the probability of finding the particle at position x. But we also need

∫

P (x′)dx′^ = 1

• the probability of finding the particle at some x is unity. The transition from sums to integrals corresponds to the transition from discrete probabilities to probability distribu- tions, and so the correct interpretation is that P (x)dx is the probability of finding the particle in an infinitesimal interval between x and x + dx.
• Since 〈x|xˆ|α〉 = x〈x|α〉 = xα(x), we can say that the position operator in the position space representation is simply x, and operating with it is equivalent to multiplying the wave function by x.
• If we make the hypothesis that 〈x|pˆ|α〉 = −iℏdα(x)/dx, we can see that the required commutation relation [ˆx, pˆ] = iℏ is satisfied.
• We can equally imagine states of definite momentum, |p〉, with ˆp|p〉 = p|p〉. Then ˜α(p) = 〈p|α〉 is the wave function in position space; in this representation ˆp is given by p and ˆx by (iℏ)d/dp.
• Let’s write the position-space wave function of |p〉, 〈x|p〉, as φp(x). Since 〈x|pˆ|p〉 can be written as either pφp(x) or −iℏdφp(x)/dx, we see that φp(x) = (1/ √ 2 πℏ) eipx/ℏ. This is just a plane wave, as expected. (The normalisation is determined by 〈p|p′〉 = δ(p − p′).)
• This implies that α(x) = (1/√ 2 πℏ)

∫

eip ′x/ℏ α˜(p′)dp′, that is the position- and momentum- space wave functions are Fourier transforms of each other. (The 1/

√

ℏ is present in the prefactor because we are using p and not k = p/ℏ as the conjugate variable.)

• Strictly speaking we should write the position-space representations of ˆx and ˆp as 〈x|ˆx|x′〉 = x′δ(x−x′) and 〈x|pˆ|x′〉 = (iℏ)dδ(x−x′)/dx′. These only make sense in a larger expression in which integration over x′^ is about to be performed.
• The extension to three dimensions is trivial, with the representation of the vector operator pˆ in position space being −iℏ∇. Since differentiation with respect to one coordinate commutes with multiplication by another, [ˆxi, pˆj ] = iℏδij as required. (We have used ˆx as the position operator. We will however use r as the position vector. The reason we don’t use ˆr is that it is so commonly used to mean a unit vector. Thus ˆx|r〉 = r|r〉.) The generalisation of φp(x) is φp(r) = (1/ √ 2 πℏ) eip·r/ℏ, which is a plane wave travelling in the direction of p.
• In the position representation, the Schr¨odinger equation reads

−

ℏ^2

2 m

∇^2 ψ(r, t) + V (r)ψ(r, t) = iℏ

∂

∂t

ψ(r, t)

Note though that position and time are treated quite differently in quantum mechanics. There is no operator corresponding to time, and t is just part of the label of the state: ψ(r, t) = 〈r|ψ(t)〉.

• Together with the probability density, ρ(r) = |ψ(r)|^2 , we also have a probability flux j(r) = (−iℏ/ 2 m)(ψ(r)∗∇ψ(r) − ψ(r)∇ψ(r)∗). The continuity equation ∇ · j = −∂ρ/∂t which ensures local conservation of probability density follows from the Schr¨odinger equation.

• A two-particle state has a wave function which is a function of the two positions, Φ(r 1 , r 2 ), and the basis kets are direct product states |r 1 〉 ⊗ |r 2 〉. For states of non-interacting distinguishable particles where it is possible to say that the first particle is in single- particle state |ψ〉 and the second in |φ〉, the state of the system is |ψ〉 ⊗ |φ〉 and the wave function is Φ(r 1 , r 2 ) = (〈r 1 | ⊗ 〈r 2 |)(|ψ〉 ⊗ |φ〉) = 〈r 1 |ψ〉〈r 2 |φ〉 = ψ(r 1 )φ(r 2 ).
• From iℏ (^) ddt |ψ(t)〉 = Ĥ |ψ(t)〉 we obtain |ψ(t + dt)〉 =

(

1 − (^) ℏi Ĥ dt

)

|ψ(t)〉. Thus

|ψ(t)〉 = lim N →∞

(

1 −

i ℏ

(t − t 0 ) N

̂

H

)N

|ψ(t 0 )〉 = e−i^ Hb(t−t 0 )/ℏ |ψ(t 0 )〉 ≡ U (t, t 0 )|ψ(t 0 )〉

where the exponential of an operator is defined as eΩb^ =

∑

n

1 n!

̂

Ω

n. If the Hamiltonian

depends explicitly on time, we have U (t, t 0 ) = T exp

(

−i

∫ (^) t t 0

̂

H(t

′)dt′/ℏ

)

, where the time- ordered exponential denoted by T exp means that in expanding the exponential, the op- erators are ordered so that Ĥ (t 1 ) always sits to the right of Ĥ (t 2 ) (so that it acts first) if t 1 < t 2. (This will be derived later, and is given here for completeness.)

References

• Shankar ch 1.
• Mandl ch 12.

1.3 The Stern-Gerlach experiment

Summary: Simple extensions of the Stern-Gerlach experiment reduce the predic- tions about quantum measurement to their essence.

The usual reason for considering the Stern-Gerlach experiment is that it shows experi- mentally that angular momentum is quantised, and that particles can have intrinsic angular momentum which is an integer multiple of 12 ℏ. An inhomogeneous magnetic field deflects par- ticles by an amount proportional to the component of their magnetic moment parallel to the field; when a beam of atoms passes through they follow only a few discrete paths (2j + 1 where j is their total angular momentum quantum number) rather than, as classically predicted, a continuum of paths (corresponding to a magnetic moment which can be at any angle relative to the field). For our purposes though a Stern-Gerlach device is a quick and easily-visualised way of making measurements of quantities (spin components) for which the corresponding operators do not commute, and thereby testing the postulates concerned with measurement. We restrict ourselves to spin-^12 ; all we need to know is that if we write | + n〉 to indicate a particle with its spin up along the direction n, and | − n〉 for spin down, then the two orthogonal states {| ± n〉} span the space for any n, and |〈±n| ± n′〉|^2 = 12 if n and n′^ are perpendicular.

In the figure above we see an unpolarised beam entering a Stern-Gerlach device with its field oriented in the z-direction. If we intercept only the upper or lower exiting beam, we have a pure spin-up or spin-down state. Here we see a sequence of Stern-Gerlach devices with their fields oriented in either the x- or z-direction. The numbers give the fraction of the original unpolarised beam which reaches this point. The sequence of two x-devices in a row illustrates reproducibility. The final z-device is the really crucial one. Do you understand why half of the particles end up spin-down, even though we initially picked the | + z〉 beam?

• Townsend ch 1.

1.4 Ehrenfest’s Theorem and the Classical Limit

Summary: The form of classical mechanics which inspired Heisenberg’s formula- tion of Classical Mechanics allows us to see when particles should behave classically.

Using iℏ (^) ddt |ψ(t)〉 = Ĥ |ψ(t)〉 and hence −iℏ (^) ddt 〈ψ(t)| = 〈ψ(t)|Ĥ , and writing 〈Ω̂ 〉 ≡ 〈ψ(t)|Ω|ψ(t)〉, we have Ehrenfest’s Theorem

d dt

〈 Ω̂ 〉 =

iℏ

〈[ Ω̂ , Ĥ ]〉 + 〈

∂ Ω̂

∂t

〉

The second term disappears if Ω is a time-independent operator (like momentum, spin...).̂ Note we are distinguishing between intrinsic time-dependence of an operator, and the time- dependence of its expectation value in a given state. This is very reminiscent of a result which follows from Hamilton’s equations in classical mechanics, for a function of position, momentum (and possibly time explicitly) Ω(p, x, t)

d dt

Ω(p, x, t) =

∂Ω

∂x

dx dt

+

∂Ω

∂p

dp dt

+

∂Ω

∂t

=

∂Ω

∂x

∂H

∂p

−

∂Ω

∂p

∂H

∂x

+

∂Ω

∂t

≡ {Ω, H} +

∂Ω

∂t

where the notation {Ω, H} is called the Poisson bracket of Ω and H, and is simply defined in terms of the expression on the line above which it replaced. (For Ω = x and Ω = p we can in fact recover Hamilton’s equations for ˙p and ˙x from this more general expression.)

In fact for Ĥ = ˆp

2 2 m +^ V^ (ˆx), we can further show that

d dt

〈xˆ〉 = 〈

pˆ m

〉 and

d dt

〈pˆ〉 = −〈

dV (ˆx) dˆx

〉

which looks very close to Newton’s laws. Note though that 〈dV (ˆx)/dˆx〉 6 = d〈V (ˆx)〉/d〈xˆ〉 in general. This correspondence is not just a coincidence, in the sense that Heisenberg was influenced by it in coming up with his formulation of quantum mechanics. It confirms that it is the expectation value of an operator, rather than the operator itself, which is closer to the classical concept of the time evolution of some quantity as a particle moves along a trajectory. Similarity of formalism is not the same as identity of concepts though. Ehrenfest’s Theorem does not say that the expectation value of a quantity follows a classical trajectory in general. What it does ensure is that if the uncertainty in the quantity is sufficiently small, in other words if ∆x and ∆p are both small (in relative terms) then the quantum motion will aproximate the classical path. Of course because of the uncertainty principle, if ∆x is small then ∆p is large, and it can only be relatively small if p itself is really large—ie if the particle’s mass is macroscopic. More specifically, we can say that we will be in the classical regime if the de Broglie wavelength is much less that the (experimental) uncertainty in x. (In the Stern-Gerlach experiment the atoms are heavy enough that (for a given component of their magnetic moment) they follow approximately classical trajectories through the inhomogeneous magnetic field.)

• Shankar ch 2.7, ch 6
• Mandl ch 3.

Approximate methods I: variational

method and WKB

It is often (almost always!) the case that we cannot solve real problems analytically. Only a very few potentials have analytic solutions, by which I mean one can write down the energy levels and wave functions in closed form, as for the harmonic oscillator and Coulomb potential. In fact those are really the only useful ones (along with square wells)... In the last century, a number of approximate methods have been developed to obtain information about systems which can’t be solved exactly. These days, this might not seem very relevant. Computers can solve differential equations very efficiently. But:

• It is always useful to have a check on numerical methods
• Even supercomputers can’t solve the equations for many interacting particle exactly in a reasonable time (where “many” may be as low as four, depending on the complexity of the interaction) — ask a nuclear physicist or quantum chemist.
• Quantum field theories are systems with infinitely many degrees of freedom. All ap- proaches to QFT must be approximate.
• If the system we are interested in is close to a soluble one, we might obtain more insight from approximate methods than from numerical ones. This is the realm of perturbation theory. The most accurate prediction ever made, for the anomalous magnetic moment of the electron, which is good to one part in 10^12 , is a 4th order perturbative calculation.

2.1 Variational methods: ground state

Summary: Whatever potential we are considering, we can always obtain an upper bound on the ground-state energy.

Suppose we know the Hamiltonian of a bound system but don’t have any idea what the energy of the ground state is, or the wave function. The variational principle states that if we simply guess the wave function, the expectation value of the Hamiltonian in that wave function will be greater than the true ground-state energy:

〈Ψ|Ĥ |Ψ〉 〈Ψ|Ψ〉

≥ E 0

This initially surprising result is more obvious if we consider expanding the (normalised)

|Ψ〉 in the true energy eigenstates |n〉, which gives 〈Ĥ 〉 =

∑

n PnEn. Since all the probabilities

10

Pn are non-negative, and all the En greater than or equal to E 0 , this is obviously not less than E 0. It is also clear that the better the guess (in the sense of maximising the overlap with the true ground state) the lower the energy bound, till successively better guesses converge on the true result. As a very simple example, consider the infinite square well with V = 0, 0 < x < a and V = ∞ otherwise. As a trial function, we use Ψ(x) = x(a − x), 0 < x < a and Ψ(x) = 0 otherwise. Then 〈Ψ|Ĥ |Ψ〉 〈Ψ|Ψ〉

=

10 ℏ^2

2 ma^2

= 1. 013

π^2 ℏ^2 2 ma^2

This is spectacularly good! Obviously it helped that our trial wave function looked a lot like what we’d expect of the true solution - symmetric about the midpoint, obeying the boundary conditions, no nodes.... In general, we will do better if we have an adjustable parameter, because then we can find the value which minimises our upper bound. So we could try Ψ(x) = x(a − x) + bx^2 (a − x)^2 (with our previous guess corresponding to b = 0). Letting Mathematica do the dirty work, we get an energy bound which is a function of b, which takes its minimum value of 1. 00001 E 0 at b = 1. 133 /a^2. Not much room for further improvement here!

Above we have plotted, on the left, the true and approximate wave functions (except that the true one is hidden under the second approximation, given in blue) and on the right, the deviations of the approximate wave functions from the true one (except that for the second approximation the deviation has been multiplied by 5 to render it visible!) This illustrates a general principle though: the wave function does have deviations from the true one on the part-per-mil scale, while the energy is good to 1 part in 10^5. This is because the error in the energy is proportional to the coefficients squared of the admixture of “wrong” states, whereas the error in the wave function is linear in them. Another example, for which the analytic solution is not known, is the quartic potential, V (x) = βx^4. Here a Gaussian trial wave function Ψ(x) = e−ax (^2) / 2 gives an upper bound for the ground energy state of 38 61 /^3 = 0.68 in units of (ℏ^4 β/m^2 )^1 /^3. (The value obtained from numerical solution of the equation is 0.668).

• Gasiorowicz ch 14.
• Mandl ch 8.

• Shankar ch 16.

2.2 Variational methods: excited states

Summary: Symmetry considerations may allow us to extend the variational method to certain excited states.

Looking again at the expression 〈Ĥ 〉 =

∑

n PnEn, and recalling that the^ Pn^ are the squares of the overlap between the trial function and the actual eigenstates of the system, we see that we can only find bounds on excited states if we can arrange for the overlap of the trial wave function with all lower states to be zero. Usually this is not possible. However an exception occurs where the states of the system can be separated into sets with different symmetry properties or other quantum numbers. Examples include parity and (in 3 dimensions) angular momentum. For example the lowest state with odd parity will automatically have zero overlap with the (even-parity) ground state, and so an upper bound can be found for it as well. For the square well, the relevant symmetry is reflection about the midpoint of the well. If we choose a trial function which is antisymmetric about the midpoint, it must have zero overlap with the true ground state. So we can get a good bound on the first excited state, since 〈Ĥ 〉 =

∑

n> 0 PnEn^ > E^1.^ Using Ψ^1 (x) =^ x(a^ −^ x)(2x^ −^ a),^0 < x < a^ we get^ E^1 ≤ 42 ℏ^2 / 2 ma^2 = 1. 064 E 1. If we wanted a bound on E 2 , we’d need a wave function which was orthogonal to both the ground state and the first excited state. The latter is easy by symmetry, but as we don’t know the exact ground state (or so we are pretending!) we can’t ensure the first. We can instead form a trial wave function which is orthogonal to the best trial ground state, but we will no longer have a strict upper bound on the energy E 2 , just a guess as to its value. In this case we can choose Ψ(x) = x(a − x) + bx^2 (a − x)^2 with a new value of b which gives orthogonality to the previous state, and then we get E 2 ∼ 10. 3 E 0 (as opposed to 9 for the actual value).

• Mandl ch 8.
• Shankar ch 16.

2.3 Variational methods: the helium atom

Summary: The most famous example of the variational principle is the ground state of the two-electron helium atom.

If we could switch off the interactions between the electrons, we would know what the ground state of the helium atom would be: Ψ(r 1 , r 2 ) = φZ 100 =2 (r 1 )φZ 100 =2 (r 2 ), where φZnlm is a single-particle wave function of the hydrogenic atom with nuclear charge Z. For the ground state n = 1 and l = m = 0 (spherical symmetry). The energy of the two electrons would be 2 Z^2 ERy = − 108 .8 eV. But the experimental energy is only − 78 .6 eV (ie it takes 78.6 eV to fully ionise neutral helium). The difference is obviously due to the fact that the electrons repel one another.

The full Hamiltonian (ignoring the motion of the proton - a good approximation for the accuracy to which we will be working) is

−

ℏ^2

2 m

(∇^21 + ∇^22 ) − 2 ℏcα

(

|r 1 |

+

|r 2 |

)

• ℏcα

|r 1 − r 2 |

where ∇^21 involves differentiation with respect to the components of r 1 , and α = e^2 /(4π 0 ℏc) ≈ 1 /137. (See here for a note on units in EM.) A really simple guess at a trial wave function for this problem would just be Ψ(r 1 , r 2 ) as written above. The expectation value of the repulsive interaction term is (5Z/4)ERy giving a total energy of − 74 .8 eV. (Gasiorowicz demonstrates the integral, as do Fitzpatrick and Branson.) It turns out we can do even better if we use the atomic number Z in the wave function Ψ as a variational parameter (that in the Hamiltonian, of course, must be left at 2). The best value turns out to be Z = 27/16 and that gives a better upper bound of − 77 .5 eV – just slightly higher than the experimental value. (Watch the sign – we get an lower bound for the ionization energy.) This effective nuclear charge of less than 2 presumably reflects the fact that to some extent each electron screens the nuclear charge from the other.

• Gasiorowicz ch 14.2, 14.4 (most complete)
• Mandl ch 7.2, 8.8.2 (clearest)
• Shankar ch 16.1 (not very clear; misprints in early printings)

2.4 WKB approximation

Summary: The WKB approximation works for potentials which are slowly- varying on the scale of the wavelength of the particle and is particularly useful for describing tunnelling.

The WKB approximation is named for G. Wentzel, H.A. Kramers, and L. Brillouin, who independently developed the method in 1926. There are pre-quantum antecedents due to Jeffreys and Raleigh, though. We can always write the one-dimensional Schr¨odinger equation as

d^2 φ dx^2

= −k(x)^2 φ(x)

where k(x) ≡

√

2 m(E − V (x))/ℏ. We could think of the quantity k(x) as a spatially-varying wavenumber (k = 2π/λ), though we anticipate that this can only make sense if it doesn’t change too quickly with position - else we can’t identify a wavelength at all. Let’s see under what conditions a solution of the form

ψ(x) = A exp

(

±i

∫ (^) x k(x′)dx′

)

might be a good approximate solution. Plugging this into the SE above, the LHS reads −(k^2 ∓ ik′)ψ. (Here and hereafter, primes denote differentiation wrt x — except when they indicate

an integration variable.) So provided |k′/k|  |k|, or |λ′|  1, this is indeed a good solution as the second term can be ignored. And |λ′|  1 does indeed mean that the wavelength is slowly varying. (One sometimes reads that what is needed is that the potential is slowly varying. But that is not a well defined statement, because dV /dx is not dimensionless. For any smooth potential, at high-enough energy we will have |λ′|  1. What is required is that the lengthscale of variation of λ, or k, or V (the scales are all approximately equal) is large compared with the de Broglie wavelength of the particle. An obvious problem with this form is that the current isn’t constant: if we calculate it we get |A|^2 ℏk(x)/m. A better approximation is

ψ(x) =

A

√

k(x)

exp

(

±i

∫ (^) x k(x′)dx′

)

which gives a constant flux. (Classically, the probability of finding a particle in a small region is inversely proportional to the speed with which it passes through that region.) Furthermore one can show that if the error in the first approximation is O |λ′|, the residual error with the second approximation is O |λ′|^2. At first glance there is a problem with the second form when k(x) = 0, ie when E = V (x). But near these points - the classical turning points - the whole approximation scheme is invalid, because λ → ∞ and so the potential cannot be “slowly varying” on the scale of λ. For a region of constant potential, of course, there is no difference between the two approx- imations and both reduce to a plain wave, since

∫ (^) x k(x′)dx′^ = kx. For regions where E < V (x), k(x) will be imaginary and there is no wavelength as such. But defining λ = 2π/k still, the WKB approximation will continue to be valid if |λ′|  1. Tunnelling and bound-state problems inevitably include regions where E ≈ V (x) and the WKB approximation isn’t valid. This would seem to be a major problem. However if such regions are short the requirement that the wave function and its derivative be continuous can help us to “bridge the gap”.

• Gasiorowicz Supplement 4 A
• Shankar ch 16.

2.4.1 WKB approximation for bound states

In a bound state problem with potential V (x), for a given energy E, we can divide space into classically allowed regions, for which E > V (x), and classically forbidden regions for which E < V (x). For simplicity we will assume that there are only three regions in total, classically forbidden for x < a and x > b, and classically allowed for a < x < b. In the classically allowed region a < x < b the wave function will be oscillating and we can write it either as a superposition of right- and left-moving complex exponentials or as

ψ(x) =

A

√

k(x)

cos

(∫ (^) x k(x′)dx′^ + φ

)

For the particular case of a well with infinite sides the solution must vanish at the boundaries, so (taking the lower limit of integration as a for definiteness; any other choice just shifts φ)

φ = (n′^ + 12 )π and

∫ (^) b a k(x

′)dx′ (^) + φ = (n′′ (^) + 1 2 )π; in other words^

∫ (^) b a k(x

′)dx′ (^) = (n + 1)π, with

integer n ≥ 0. Of course for constant k this gives k = nπ/(b − a), which is exact.

For a more general potential, outside the classically allowed region we will have decaying exponentials. In the vicinity of the turning points these solutions will not be valid, but if we approximate the potential as linear we can solve the Schr¨odinger equation exactly (in terms of Airy functions). Matching these to our WKB solutions in the vicinity of x = a and x = b gives the surprisingly simple result that inside the well

ψ(x) =

A

√

k(x)

cos

(∫ (^) x

a

k(x′)dx′^ − π/ 4

)

and ψ(x) =

A′

√

k(x)

cos

(∫ (^) x

b

k(x′)dx′^ + π/ 4

)

which can only be satisfied if A′^ = ±A and

∫ (^) b a k(x

′)dx′ (^) = (n+ 1 2 )π. This latter is the quantisation condition for a finite well; it is different from the infinite well because the solution can leak into the forbidden region. (For a semi-infinite well, the condition is that the integral equal (n + 34 )π. This is the appropriate form for the l = 0 solutions of a spherically symmetric well.) Unfortunately we can’t check this against the finite square well though, because there the potential is definitely not slowly varying at the edges, nor can it be approximated as linear. But we can try the harmonic oscillator, for which the integral gives Eπ/ℏω and hence the quantisation condition gives E = (n + 12 ) ℏω! The approximation was only valid for large n (small wavelength) but in fact we’ve obtained the exact answer for all levels. Details of the matching process are given in section 2.4.1.1, since I’ve not found them in full detail in any textbook. They are not examinable.

• Gasiorowicz Supplement 4 A
• Shankar ch 16.

2.4.1.1 Matching with Airy Functions

This section is not examinable. More about Airy functions can be found in section A.7. If we can treat the potential as linear over a wide-enough region around the turning points that, at the edges of the region, the WKB approximation is valid, then we can match the WKB and exact solutions. Consider a linear potential V = βx as an approximation to the potential near the right-hand turning point b. We will scale x = ( ℏ^2 /(2mβ))^1 /^3 z and E = (ℏ^2 β^2 / 2 m)^1 /^3 μ, so the turning point is at z = μ. Then the differential equation is y′′(z) − zy(z) + μy(z) = 0 and the solution which decays as z → ∞ is y(z) = AAi(z − μ). This has to be matched, for z not too close to μ, to the WKB solution. In these units, k(x) = (μ−z) and

∫ (^) x b k(x

′)dx′ (^) = ∫^ z μ (μ−z

′)dz′ (^) = −(2/3)(μ−z) 3 / (^2) ,

so

ψWKB x<b (z) =

B

(μ − z)^1 /^4

cos

(

−^23 (μ − z)^3 /^2 + φ

)

and ψWKB x>b (z) =

C

(z − μ)^1 /^4

exp

(

−^23 (z − μ)^3 /^2

)

.

(We chose the lower limit of integration to be μ in order that the constant of integration vanished; any other choice would just shift φ.) Now the asymptotic forms of the Airy function are known:

Ai(z) z→−∞ −→

cos

(

2 3 |z|

3 / (^2) − π 4

)

√

π |z|^1 /^4

and Ai(z) z→∞ −→

e−^

(^23) z 3 / 2

2

√

πz^1 /^4

so

Ai(z − μ) z→−∞ −→

cos

( 2

3 (μ^ −^ z)

3 / (^2) − π 4

)

√

π(μ − z)^1 /^4

and Ai(z − μ) z→∞ −→

e−^

(^23) (z−μ) 3 / 2

2

√

π(z − μ)^1 /^4

and these will match the WKB expressions exactly provided C = 2B and φ = π/4. At the left-hand turning point a, the potential is V = −βx (with a different β in general) and the solution y(z) = AAi(−z − μ). On the other hand the WKB integral is

∫ (^) x a k(x

′)dx′ (^) = ∫ (^) z μ (μ^ +^ z

′)dz′ (^) = 2/3(μ + z) 3 / (^2). So in the classically allowed region we are comparing

Ai(−z−μ) z→∞ −→

cos

( 2

3 (z^ +^ μ)

3 / (^2) − π 4

)

√

π(z + μ)^1 /^4

with ψWKB x>a (z) =

D

(μ + z)^1 /^4

cos

( 2

3 (μ^ +^ z)

3 / (^2) + φ)

which requires φ = −π/4. (Note that φ is different in each case because we have taken the integral from a different place.) It is worth stressing that though the exact (Airy function) and WKB solutions match “far away” from the turning point, they do not do so close in. The (z − μ)−^1 /^4 terms in the latter mean that they blow up, but the former are perfectly smooth. They are shown (for μ = 0) below, in red for the WKB and black for the exact functions. We can see they match very well so long as |z − μ| > 1; in fact z → ∞ is overkill!

So now we can be more precise about the conditions under which the matching is possible: we need the potential to be linear over the region ∆x ∼ ( ℏ^2 /(2mβ))^1 /^3 where β = dV /dx. Linearity means that ∆V /∆x ≈ dV /dx at the turning point, or

∣

∣d^2 V /dx^2

∣

∣ (^) ∆x  dV /dx

(assuming the curvature is the dominant non-linearity, as is likely if V is smooth). For the harmonic oscillator,

∣

∣d^2 V /dx^2

∣

∣ (^) ∆x/(dV /dx) = 2(ℏω/E)^2 /^3 which is only much less than 1 for

very large n, making the exact result even more surprising!

2.4.2 WKB approximation for tunnelling

For the WKB approximation to be applicable to tunnelling through a barrier, we need as always |λ′|  1. In practice that means that the barrier function is reasonably smooth and that E  V (x). Now it would of course be possible to do a careful calculation, writing down the WKB wave function in the three regions (left of the barrier, under the barrier and right of the barrier), linearising in the vicinity of the turning points in order to match the wave function and its derivatives at both sides. This however is a tiresomely lengthy task, and we will not attempt it. Instead, recall the result for a high, wide square barrier; the transmission coefficient in the limit e−^2 κ∆L^  1 is given by

T =

16 k 1 k 2 κ^2 (κ^2 + k^21 )(κ^2 + k 22 )

e−^2 κL

where√ k 1 and k 2 are the wavenumbers on either side of the barrier (width L, height V ) and κ = 2 m(V − E). (See the notes for PHYS20101, where however k 1 = k 2 .) All the prefactors are

not negligible, but they are weakly energy-dependent, whereas the e−^2 κL^ term is very strongly energy dependent. If we plot log T against energy, the form will be essentially const − 2 κ(E)L, and so we can still make predictions without worrying about the constant. For a barrier which is not constant, the WKB approximation will yield a similar expression for the tunnelling probability:

T = [prefactor] × exp(− 2

∫ (^) b

a

κ(x′)dx′)

where κ(x) ≡

√

2 m(V (x) − E)/ℏ. The WKB approximation is like treating a non-square barrier like a succession of square barriers of different heights. The need for V (x) to be slowly varying is then due to the fact that we are slicing the barrier sufficiently thickly that e−^2 κ∆L^  1 for each slice. The classic application of the WKB approach to tunnelling is alpha decay. The potential here is a combination of an attractive short-range nuclear force and the repulsive Coulomb interaction between the alpha particle and the daughter nucleus. Unstable states have energies greater than zero, but they are long-lived because they are classically confined by the barrier. (It takes some thought to see that a repulsive force can cause quasi-bound states to exist!) The semi-classical model is of a pre-formed alpha particle bouncing back and forth many times (f ) per second, with a probability of escape each time given by the tunnelling probability, so the decay rate is given by 1/τ = f T. Since we can’t calculate f with any reliability we would be silly to worry about the prefactor in T , but the primary dependence of the decay rate on the energy of the emitted particle will come from the easily-calculated exponential.

In the figure above the value of a is roughly the nuclear radius R, and b is given by VC (b) = E, with the Coulomb potential VC (r) = zZ ℏcα/r. (Z is the atomic number of the daughter nucleus and z = 2 that of the alpha.) The integral in the exponent can be done (see Gasiorowicz Supplement 4 B for details; the substitution r = b cos^2 θ is used), giving in the limit b  a

∫ (^) b

a

κ(x′)dx′^ = 2πzZα

√

mc^2 2 E

= 39

Z

√

E(MeV)

⇒ log 10

τ

= const − 1. 72

Z

√

E(MeV)

Data for the lifetimes of long-lived isotopes (those with low-energy alphas) fit such a functional form well, but with 1.61 rather than 1.72. In view of the fairly crude approximations made, this is a pretty good result. Note it is independent of the nuclear radius because we used b  a; we could have kept the first correction, proportional to

√

b/a, to improve the result.

• Gasiorowicz Supplement 4 B
• Shankar ch 16.

Approximate methods II:

Time-independent perturbation theory

3.1 Formalism

Summary: Perturbation theory is the most widely used approximate method. “Time-independent perturbation theory” deals with bound states eg the spectrum of the real hydrogen atom and its response to extermal fields.

Perturbation theory is applicable when the Hamiltonian Ĥ can be split into two parts, with the first part being exactly solvable and the second part being small in comparison. The first part is always written Ĥ (0), and we will denote its eigenstates by |n(0)〉 and energies by E n(0) (with wave functions φ(0) n ). These we know. The eigenstates and energies of the full Hamiltonian are denoted |n〉 and En, and the aim is to find successively better approximations to these. The zeroth-order approximation is simply |n〉 = |n(0)〉 and En = E n(0) , which is just another way of saying that the perturbation is small. Nomenclature for the perturbing Hamiltonian Ĥ − Ĥ (0)^ varies. δV , Ĥ (1)^ and λĤ (1)^ are all common. It usually is a perturbing potential but we won’t assume so here, so we won’t use the first. The second and third differ in that the third has explicitly identified a small, dimensionless parameter (eg α in EM), so that the residual Ĥ (1)^ isn’t itself small. With the last choice, our expressions for the eigenstates and energies of the full Hamiltonian will be explicitly power series in λ, so En = E n(0) + λE n(1) + λ^2 E n(2) +... etc. With the second choice the small factor is

hidden in Ĥ (1), and is implicit in the expansion which then reads En = E n(0) +E n(1) +E n(2) +.. .. In this case one has to remember that anything with a superscript (1) is first order in this implicit small factor, or more generally the superscript (m) denotes something which is mth order. For the derivation of the equations we will retain an explicit λ, but thereafter we will set it equal to one to revert to the other formulation. We will take λ to be real so that Ĥ 1 is Hermitian. We start with the master equation

( Ĥ (0)^ + λ Ĥ (1))|n〉 = En|n〉.

Then we substitute in En = E n(0) + λE n(1) + λ^2 E n(2) +... and |n〉 = |n(0)〉 + λ|n(1)〉 + λ^2 |n(2)〉 +... and expand. Then since λ is a free parameter, we have to match terms on each side with the same powers of λ, to get

Ĥ (0)|n(0)〉 = E n(0) |n(0)〉 Ĥ (0)|n(1)〉 + Ĥ (1)|n(0)〉 = E n(0) |n(1)〉 + E n(1) |n(0)〉 Ĥ (0)|n(2)〉 + Ĥ (1)|n(1)〉 = E n(0) |n(2)〉 + E n(1) |n(1)〉 + E n(2) |n(0)〉

We have to solve these sequentially. The first we assume we have already done. The second will yield E n(1) and |n(1)〉. Once we know these, we can use the third equation to yield E(2) n and |n(2)〉, and so on. In each case, to solve for the energy we take the inner product with 〈n(0)| (ie the same state)

whereas for the wave function, we use 〈m(0)| (another state). We use, of course, 〈m(0)|Ĥ (0)^ = E m(0) 〈m(0)| and 〈m(0)|n(0)〉 = δmn. At first order we get

E n(1) = 〈n(0)|Ĥ (1)|n(0)〉 and 〈m(0)|n(1)〉 =

〈m(0)|Ĥ (1)|n(0)〉 E(0) n − E(0) m

∀m 6 = n

The second equation tells us the overlap of |n(1)〉 with all the other |m(0)〉, but not with |n(0)〉. This is obviously not constrained, because we can add any amount of |n(0)〉 and the equations will still be satisfied. However we need the state to continue to be normalised, and when we expand 〈n|n〉 = 1 in powers of λ we find that 〈n(0)|n(1)〉 is required to be imaginary. Since this is just like a phase rotation of the original state and we can ignore it. Hence

|n(1)〉 =

∑

m 6 =n

〈m(0)|Ĥ (1)|n(0)〉 E n(0) − E m(0)

|m(0)〉

If the spectrum of Ĥ (0)^ is degenerate, there is a potential problem with this expression because the denominator can be infinite. In that case we have to diagonalise Ĥ (1)^ in the subspace of degenerate states exactly. This is called “degenerate perturbation theory”. Then at second order

E n(2) = 〈n(0)|Ĥ (1)|n(1)〉 =

∑

m 6 =n

∣

∣

∣〈m(0)|Ĥ (1)|n(0)〉

∣

∣

∣

2

E n(0) − E m(0)

The expression for the second-order shift in the wave function |n(2)〉 can also be found but it is tedious. The main reason we wanted |n(1)〉 was to find E n(2) anyway, and we’re not planning to find E n(3)! Note that though the expression for E(1) n is generally applicable, those for |n(1)〉 and E n(2) would need some modification if the Hamiltonian had continuum eigenstates as well as bound states (eg hydrogen atom). Provided the state |n〉 is bound, that is just a matter of integrating rather than summing. This restriction to bound states is why Mandl calls chapter 7 “bound-state perturbation theory”. The perturbation of continuum states (eg scattering states) is usually dealt with separately. Note that the equations above hold whether we have identified an explicit small parameter λ or not. So from now on we will set λ to one, and En = E n(0) + E(1) n + E n(2) +.. .. Connection to variational approach: For the ground state (which is always non-degenerate) E 0 (0) + E 0 (1) is a variational upper bound on the exact energy E 0 , since it is obtained by using the unperturbed ground state as a trial wavefunction for the full Hamiltonian. It follows that the sum of all higher corrections E 0 (2) +... must be negative. We can see indeed that E 0 (2) will always be negative, since for every term in the sum the numerator is positive and the denominator negative.

• Gasiorowicz ch 11.1
• Mandl ch 7.1,3
• Shankar ch 17.1
• Townsend ch 11.1-3

3.1.1 Simple examples of perturbation theory

Probably the simplest example we can think of is an infinite square well with a low step half way across, so that V (x) = 0 for 0 < x < a/2, V 0 for a/ 2 < x < a and infinite elsewhere. We treat this as a perturbation on the flat-bottomed well, so H(1)^ = V 0 for a/ 2 < x < a and zero elsewhere.

The ground-state unperturbed wavefunction is ψ 0 (0) =

√

2 a sin^

πx a , with unperturbed energy E 0 (0) = π^2 ℏ^2 /(2ma^2 ). A “low” step will mean V 0  E 0 (0). Then we have

E 0 (1) = 〈ψ(0) 0 |H(1)|ψ(0) 0 〉 =

a

∫ (^) a

a/ 2

V 0 sin^2

πx a

dx =

V 0

This problem can be solved semi-analytically; in both regions the solutions are sinusoids, but with wavenumbers k =

√

2 mE/ℏ and k′^ =

√

2 m(E − V 0 )/ℏ respectively; satisfying the bound- ary conditions and matching the wavefunctions and derivatives at x = a/2 gives the condition k cot(ka/2) = k′^ cot(k′a/2) which can be solved numerically for E. (You ought to try this, it will be good practice for later sections of the course.) Below the exact solution (green, dotted) and E 0 (0) + E(1) 0 (blue) are plotted; we can see that they start to diverge when V 0 = 5 (everything is in units of ℏ^2 /(2ma^2 )).

We can also plot the exact wavefunctions for different step size, and see that for V 0 = 10 (the middle picture, well beyond the validity of first-order perturbation theory) it is significantly different from a simple sinusoid.

Another example is the harmonic oscillator, with a perturbing potential H(1)^ = λx^2. The states of the unperturbed oscillator are denoted |n(0)〉 with energies E 0 (0) = (n + 12 )ℏω.

√ Recalling that in terms of creation and annihilation operators (see section A.4),^ xˆ^ = ℏ/(2mω)(ˆa + ˆa†), with [ˆa, ˆa†] = 1, and so

E n(1) = 〈n(0)|H(1)|n(0)〉 =

ℏλ 2 mω

〈n(0)|(ˆa†)^2 + ˆa^2 + 2ˆa†^ ˆa + 1|n(0)〉 =

λ mω^2

ℏω(n + 12 )

The first-order change in the wavefunction is also easy to compute, as 〈m(0)|H(1)|n(0)〉 = 0 unless m = n ± 2. Thus

|n(1)〉 =

∑

m 6 =n

〈m(0)|Ĥ (1)|n(0)〉 E(0) n − E(0) m

|m(0)〉

=

ℏλ 2 mω

(√

(n + 1)(n + 2) − 2 ℏω

|(n + 2)(0)〉 +

√

n(n − 1) 2 ℏω

|(n − 2)(0)〉

)

and E(2) n = 〈n(0)|Ĥ (1)|n(1)〉 =

∑

m 6 =n

∣

∣

∣〈m(0)|Ĥ (1)|n(0)〉

∣

∣

∣

2

E n(0) − E m(0)

=

(

ℏλ 2 mω

) 2 (

(n + 1)(n + 2) − 2 ℏω

+

n(n − 1) 2 ℏω

)

= −^12

(

λ mω^2

) 2

ℏω(n + 12 )

We can see a pattern emerging, and of course this is actually a soluble problem, as all that the perturbation has done is change the frequency. Defining ω′^ = ω

√

1 + 2λ/(mω^2 ), we see that the exact solution is

En = (n + 12 )ℏω′^ = (n + 12 )ℏω

(

1 +

λ mω^2

−

(

λ mω^2

) 2

+...

)

in agreement with the perturbative calculation.

3.2 Example of degenerate perturbation theory

Suppose we have a three state basis and an Ĥ (0)^ whose eigenstates, | 1 (0)〉, | 2 (0)〉 and | 3 (0)〉, have energies E(0) 1 , E 2 (0) and E 3 (0) (all initially assumed to be different). A representation of this system is

| 1 (0)〉 =





0

0



 , | 2 (0)〉 =





0

0



 , | 3 (0)〉 =





0

0



 , Ĥ (0)^ =





E(0) 1 0 0

0 E 2 (0) 0

0 0 E 3 (0)



 .

Now let’s consider the perturbation

Ĥ (1)^ = a







 .

Then we can show that, to first order in a

E(1) 1 = E 2 (1) = E 3 (1) = a, | 1 (1)〉 =

a E 1 (0) − E 2 (0)

| 2 (0)〉, | 2 (1)〉 =

a E(0) 2 − E(0) 1

| 1 (0)〉, | 3 (1)〉 = 0

And hence also

E(2) 1 = −E 2 (2) =

a^2 E 1 (0) − E 2 (0)

, E 3 (2) = 0

We note that because Ĥ (1)^ is already diagonal in the | 3 (0)〉 space, the first-order shift in energy is exact and there is no change to that eigenvector. In this case it is straightforward to obtain the eigenvalues of Ĥ (0)^ + Ĥ (1)^ exactly:

E 1 = (^12)

(

2 a + E 1 (0) + E(0) 2 −

√

4 a^2 + (E 2 (0) − E 1 (0) )^2

)

, E 2 = 12

(

2 a + E 1 (0) + E 2 (0) +

√

4 a^2 + (E 2 (0) − E 1 (0) )^2

)

and E 3 = E(0) 3 + a, and so we can check the expansion to order a^2. Now consider the case where E 2 (0) = E(0) 1. We note that | 1 (0)〉 and | 2 (0)〉 are just two of an infinite set of eigenstates with the same energy E 1 (0) , since any linear combination of them is another eigenstate. Our results for the third state are unchanged, but none of those obtained for the first two still hold. Instead we have to work in a new basis, | 1 ′(0)〉 and | 2 ′(0)〉 which diagonalises Ĥ (1). By inspection we see that this is

| 1 ′(0) 〉 =

√

(| 1 (0)〉 + | 2 (0)〉) and | 2 ′(0) 〉 =

√

(| 1 (0)〉 − | 2 (0)〉).

Then Ĥ (1)| 1 ′(0)〉 = 2a| 1 ′(0)〉 and Ĥ (1)| 2 ′(0)〉 = 0. Hence

E(1) 1 ′ = 2a, E 2 (1)′ = 0, | 1 ′(1) 〉 = | 2 ′(1) 〉 = 0, E 1 (2)′ = E 2 (2)′ = 0

In this case because Ĥ (1)^ doesn’t mix states 1 & 2 with 3, diagonalising it in the subspace is actually equivalent to solving the problem exactly. We can check our results against the exact eigenvalues for E 2 (0) = E 1 (0) and see that they are correct, except that we made the “wrong” choice for our labelling of | 1 ′(0)〉 and | 2 ′(0)〉.

3.3 The fine structure of hydrogen

Although the Schr¨odinger equation with a Coulomb potential reproduces the Bohr model and gives an excellent approximation to the energy levels of hydrogen, the true spectrum was known to be more complicated right from the start. The small deviations are termed “fine structure” and they are of order 10−^4 compared with the ground-state energy (though the equivalent terms for many-electron atoms can be sizable). Hence perturbation theory is an excellent framework in which to consider them. There are two effects to be considered. One arises from the use of the non-relativistic expression√ p^2 / 2 m for the kinetic energy, which is only the first term in an expansion of (mc^2 )^2 + (pc)^2 − mc^2. The first correction term is −p^4 /(8m^3 c^2 ), and its matrix elements

are most easily calculated using the trick of writing it as − 1 /(2mc^2 )(Ĥ (0)^ − VC (r))^2 , where Ĥ (0) is the usual Hamiltonian with a Coulomb potential. Now in principle we need to be careful here, because Ĥ (0)^ is highly degenerate (energies depend only on n and not on l or m). However

we have 〈nl′m′|( Ĥ (0)^ − VC (r))^2 |nlm〉 = 〈nl′m′|(E n(0) − VC (r))^2 |nlm〉, and since in this form the operator is spherically symmetric, it can’t link states of different l or m. So the basis {|nlm〉}

already diagonalises Ĥ (0)^ in each subspace of states with the same n, and we have no extra work to do here. (We are omitting the superscript (0)^ on the hydrogenic states, here and below.) The final result for the kinetic energy effect is

〈nlm|Ĥ KE(1)|nlm〉 = −

α^2 |E n(0) | n

(

2 l + 1

−

4 n

)

In calculating this the expressions E n(0) = (^21) n 2 α^2 mc^2 and a 0 = ℏ/(mcα) are useful. Tricks for doing the the radial integrals are explained in Shankar qu. 17.3.4; they are tabulated in section A.3. Details of the algebra for this and the following calculation are given here. The second correction is the spin-orbit interaction:

Ĥ (1)

SO =^

2 m^2 c^2 r

dVC dr

̂

L · ̂S

In this expression ̂L and ̂S are the vector operators for orbital and spin angular momentum respectively. The usual (somewhat hand-waving) derivation talks of the electron seeing a mag- netic field from the proton which appears to orbit it; the magnetic moment of the electron then prefers to be aligned with this field. This gives an expression which is too large by a factor of 2; an exact derivation requires the Dirac equation. This time we will run into trouble with the degeneracy of Ĥ (0)^ unless we do some work first. The usual trick of writing 2L̂ · ̂S = ̂J^2 − L̂^2 −̂ S^2 where ̂J = L̂ + Ŝ tells us that rather than working with eigenstates of L̂^2 , Lˆz , Ŝ^2 and Sˆz , which would be the basis we’d get with the minimal direct product of the spatial state |nlml〉 and a spinor |sms〉, we want to use eigenstates of L̂^2 , ̂S^2 ̂J^2 and Jˆz , |nljmj 〉, instead. (Since S = 1 2 for an electron we suppress it in the labelling

of the state.) An example of such a state is |n 11212 〉 =

√

2 3 |n^11 〉 ⊗ |

1 2 −^

1 2 〉 −

√

1 3 |n^10 〉 ⊗ |

1 2

1 2 〉. Then 〈nljmj |Ĥ SO(1)|nljmj 〉 =

α^2 |E n(0) | n

(

2 l + 1

−

2 j + 1

)

(This expression is only correct for l 6 = 0. However there is another separate effect, the Darwin term, which only affects s-waves and whose expectation value is just the same as above (with l = 0 and j = 12 ), so we can use this for all l. The Darwin term can only be understood in the context of the Dirac equation.) So finally E(1) nj =

α^2 |E n(0) | n

(

4 n

−

2 j + 1

)

.

The degeneracy of all states with the same n has been broken. States of l = j ± 12 are still degenerate, a result that persists to all orders in the Dirac equation (where in any case orbital angular momentum is no longer a good quantum number.) So the eight n = 2 states are split by 4. 5 × 10 −^5 eV, with the 2 p 3 / 2 state lying higher that the degerate 2 p 1 / 2 and 2 s 1 / 2 states. Two other effects should be mentioned here. One is the hyperfine splitting. The proton has a magnetic moment, and the energy of the atom depends on whether the electon spin is aligned with it or not— more precisely, whether the total spin of the electon and proton is 0 or

1. The anti-aligned case has lower energy (since the charges are opposite), and the splitting for the 1s state is 5. 9 × 10 −^6 eV. (It is around a factor of 10 smaller for any of the n = 2 states.) Transitions between the two hyperfine states of 1s hydrogen give rise to the 21 cm microwave radiation which is a signal of cold hydrogen gas in the galaxy and beyond. The final effect is called the Lamb shift. It cannot be accounted for in quantum mechanics, but only in quantum field theory.

The diagrams above show corrections to the simple Coulomb force which would be rep- resented by the exchange of a single photon between the proton and the electron. The most notable effect on the spectrum of hydrogen is to lift the remaining degeneracy between the 2 p 1 / 2 and 2 s 1 / 2 states, so that the latter is higher by 4. 4 × 10 −^6 eV. Below the various corrections to the energy levels of hydrogen are shown schematically. The gap between the n = 1 and n = 2 shells is supressed, and the Lamb and hyperfine shifts are exaggerated in comparison with the fine-structure. The effect of the last two on the 2 p 3 / 2 level is not shown.