Quantum Mechanics: Projection Probabilities and Density Matrix, Study notes of Quantum Mechanics

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Problems in Quantum Mechanics

Daniel F. Styer

July 1994

Contents

• 1 Stern-Gerlach Analyzers
• 2 Photon Polarization
• 3 Matrix Mathematics
• 4 The Density Matrix
• 5 Neutral K Mesons
• 6 Continuum Systems
• 7 The Free Particle
• 8 Potential Problems
• 9 Perturbation Theory for the Energy Eigenproblem
• 10 Perturbation Theory for the Time Development Problem
• 11 Angular Momentum
• 12 Central Force Motion
• 13 Identical Particles
• 14 Conclusion

2 CHAPTER 1. STERN-GERLACH ANALYZERS

1.2 Multiple analyzers An atom of state |z+〉 is shot into the following line of three Stern-Gerlach analyzers.

γ

β

|z+>

α

• or -

What is the probability that it emerges from the + output of analyzer C? From the − output? Why don’t these probabilities sum to one?

1.3 Quantum mechanics is not statistical mechanics Find the projection probabilities from state |z+〉 to states | 30 ◦+〉, | 30 ◦−〉, |x+〉, and |x−〉. Find the projection probabilities from states | 30 ◦+〉 and | 30 ◦−〉 to |x−〉. Denote the projection probability from |A〉 to |B〉 by P P (|A〉, |B〉). If we use the phrase “the probability that a system in state |A〉 is in state |B〉” for the more precise phrase “the projection probability from |A〉 to |B〉”, then it seems reasonable that

P P (|z+〉, |x−〉) = P P (|z+〉, | 30 ◦+〉) P P (| 30 ◦+〉, |x−〉) + P P (|z+〉, | 30 ◦−〉) P P (| 30 ◦−〉, |x−〉).

Use your numerical results to show that this expectation is wrong.

1.4 What is a basis state? Mr. van Dam claims that a silver atom has three, not two, basic magnetic dipole states. To back up his claim, he has constructed the following “Stern-Gerlach-van Dam” analyzer out of a z Stern-Gerlach analyzer and an x Stern-Gerlach analyzer.

Stern, Gerlanch, and van Dam, Inc.

A

B

C

(The output of the x analyzer is piped to output holes A and B using atomic pipes that do not affect the magnetic dipole state.) Show that the set of exit states {|A〉, |B〉, |C〉} is complete, but that |B〉 is not orthogonal to |C〉.

1.5 Analyzer loop

|z+>

θ a

b

?

Atoms in state |z+〉 are injected into an analyzer loop tilted an angle θ to the z direction. The output atoms are then fed into a z Stern-Gerlach analyzer. What is the probability of the atom leaving the + channel of the last analyzer when:

a. Branches a and b are both open? b. Branch b is closed? c. Branch a is closed?

Chapter 2

Photon Polarization

In lecture I have developed the principles of quantum mechanics using a particular system, the magnetic moment of a silver atom (a so-called “spin- 12 ” system), which has two basis states. Another system with two basis states is polarized light. I did not use this system mainly because photons are less familiar than atoms. This chapter develops the quantum mechanics of photon polarization much as the lectures developed the quantum mechanics of spin- 12.

One cautionary note: There is always a tendency to view the photon as a little bundle of electric and magnetic fields, a “wave packet” made up of these familiar vectors. This view is completely incorrect. In quantum electrodynamics, in fact, the electric field is a classical macroscopic quantity that takes on meaning only when a large number of photons are present.

6 CHAPTER 2. PHOTON POLARIZATION

2.1 Classical description of polarized light

unpolarized light x -polarized light (^) θ-polarized light

θ

When a beam of unpolarized light passes through a sheet of perfect polarizing material (called “Po- laroid” and represented by the symbol

l

where the arrow shows the polarizing axis), the emerging beam is of lower intensity and it is “polarized”, i.e. the electric field vector undulates but points only parallel or antiparallel to the polarizing axis. When a beam of vertically polarized light (an “x-polarized beam”) is passed through a sheet of ideal Polaroid with polarizing axis oriented at an angle θ to the vertical, the beam is reduced in intensity and emerges with an electric field undulating parallel to the sheet’s polarizing axis (a “θ-polarized beam”). Polaroid sheet performs these feats by absorbing any component of electric field perpendicular to its polarizing axis. Show that if the incoming x-polarized beam has intensity I 0 , then the outgoing θ- polarized beam has intensity I 0 cos^2 θ. Show that this expression gives the proper results when θ is 0◦, 90 ◦, 180◦^ or 270◦.

8 CHAPTER 2. PHOTON POLARIZATION

2.4 Circular polarization Just as it is possible to analyze any light beam into x- and y-polarized beams, or θ- and θ + 90◦- polarized beams, so it is possible to analyze and beam into right- and left-circularly polarized beams. You might remember from classical optics that any linearly polarized beam splits half-and-half into right- and left-circularly polarized light when so analyzed.

RL analyzer

linearly polarized light -

• (^) right-circularly polarized light
• (^) left-circularly polarized light

Quantum mechanics maintains that right- and left-circularly polarized beams are made up of photons in the |R〉 and |L〉 states, respectively. The projection amplitudes thus have magnitudes

|〈R|`p〉| = 1 /

|〈L|`p〉| = 1 /

where |`p〉 is any linearly polarized state. By building an RL analyzer loop you can convince yourself that 〈θ|R〉〈R|x〉 + 〈θ|L〉〈L|x〉 = 〈θ|x〉 = cos θ. (2.3) Show that no real valued projection amplitudes can satisfy both relations (2.2) and (2.3), but that the complex values 〈L|θ〉 = eiθ^ /

2 〈L|x〉 = 1 /

〈R|θ〉 = e−iθ^ /

2 〈R|x〉 = 1 /

are satisfactory!

Chapter 3

Matrix Mathematics

3.1 The trace For any N × N matrix A (with components aij ) the trace of A is defined by

tr(A) =

∑^ N

i=

aii

Show that tr(AB) = tr(BA), and hence that tr(ABCD) = tr(DABC) = tr(CDAB), etc. (the so- called “cyclic invariance” of the trace). However, show that tr(ABC) does not generally equal tr(CBA) by constructing a counterexample. (Assume all matrices to be square.)

3.2 The outer product Any two complex N -tuples can be multiplied to form an N × N matrix as follows: (The star represents complex conjugation.) x = (x 1 x 2... xN ) y = (y 1 y 2... yN )

x ⊗ y =

x 1 x 2 .. . xN

(y∗ 1 y∗ 2... y∗ N ) =

x 1 y∗ 1 x 1 y∗ 2... x 1 y∗ N x 2 y∗ 1 x 2 y∗ 2... x 2 y∗ N .. . xN y∗ 1 xN y 2 ∗... xN y∗ N

This so-called “outer product” is quite different from the familiar “dot product” or “inner product”

x · y = (x∗ 1 x∗ 2... x∗ N )

y 1 y 2 .. . yN

= x∗ 1 y 1 + x∗ 2 y 2 +... + x∗ N yN.

Write a formula for the i, j component of x ⊗ y and use it to show that tr(y ⊗ x) = x · y.

9

3.4 More on Pauli matrices

a. Find the eigenvalues and corresponding (normalized) eigenvectors for all three Pauli matrices. b. Define exponentiation of matrices via

eM^ =

∑^ ∞

n=

M n n!

Show that eσi^ = cosh(1)I + sinh(1)σi for i = 1, 2 , 3 and that e(σ^1 +σ^3 )^ = cosh(

2)I +

√^1

sinh(

2)(σ 1 + σ 3 ).

(Hint: Look up the series expansions of sinh and cosh.) c. Prove that eσ^1 eσ^3 6 = e(σ^1 +σ^3 ).

3.5 Hermitian operators

a. Show that if Aˆ is a linear operator and (a, Aaˆ) is real for all vectors a, then Aˆ is Hermitian. (Hint: Employ the hypothesis with a = b + c and a = b + ic.) b. Show that any operator of the form

Aˆ = ca|a〉〈a| + cb|b〉〈b| + · · · + cz |z〉〈z|,

where the cn are real constants, is Hermitian. c. You know that if an operator is Hermitian then all of its eigenvalues are real. Show that the converse is false by producing a counterexample. (Hint: Try a 2 × 2 upper triangular matrix.)

3.6 Unitary operators Show that all the eigenvalues of a unitary operator have square modulus unity.

3.7 Commutator algebra Prove that

[ A, bˆ Bˆ + c Cˆ] = b[ A,ˆ Bˆ] + c[ A,ˆ Cˆ] [a Aˆ + b B,ˆ Cˆ] = a[ A,ˆ Cˆ] + b[ B,ˆ Cˆ] [ A,ˆ Bˆ Cˆ] = Bˆ[ A,ˆ Cˆ] + [ A,ˆ Bˆ] Cˆ [ Aˆ B,ˆ Cˆ] = Aˆ[ B,ˆ Cˆ] + [ A,ˆ Cˆ] Bˆ [ A,ˆ [ B,ˆ Cˆ]] + [ C,ˆ [ A,ˆ Bˆ]] + [ B,ˆ [ C,ˆ Aˆ]] = 0 (the “Jacobi identity”).

Chapter 4

The Density Matrix

4.1 Definition Consider a system in quantum state |ψ〉. Define the operator ρˆ = |ψ〉〈ψ|, called the density matrix , and show that the expectation value of the observable associated with operator Aˆ in |ψ〉 is tr{ρˆ Aˆ}. 4.2 Statistical mechanics Frequently physicists don’t know exactly which quantum state their system is in. (For example, silver atoms coming out of an oven are in states of definite μ projection, but there is no way to know which state any given atom is in.) In this case there are two different sources of measurement uncertainty: first, we don’t know what state they system is in (statistical uncertainty, due to our ignorance) and second, even if we did know, we couldn’t predict the result of every measurement (quantum uncertainty, due to the way the world works). The density matrix formalism neatly handles both kinds of uncertainty at once. If the system could be in any of the states |a〉, |b〉,... , |i〉,... (not necessarily a basis set), and if it has probability pi of being in state |i〉, then the density matrix ρˆ =

i

pi|i〉〈i||

is associated with the system. Show that the expectation value of the observable associated with Aˆ is still given by tr{ρˆ Aˆ}. 4.3 A one-line proof (if you see the right way to do it) Show that tr{ρˆ} = 1.

14 CHAPTER 5. NEUTRAL K MESONS

between this matrix and the Pauli matrices? Show that the normalized eigenstates of CP are

|KU 〉 =

(|K^0 〉 + | K¯^0 〉),

|KS 〉 = √^1

(|K^0 〉 − | K¯^0 〉).

(The U and S stand for unstable and stable, but that’s again irrelevant because we’ll ignore K meson decay.)

5.3 The Hamiltonian The time evolution of a neutral K meson is governed by the “weak interaction” Hamiltonian

Hˆ = eˆ1 + f CP .̂

(There is no way for you to derive this. I’m just telling you.) Show that the numbers e and f must be real.

5.4 Time evolution Neutral K mesons are produced in states of definite strangeness because they are produced by the “strong interaction” Hamiltonian that conserves strangeness. Suppose one is produced at time t = 0 in state |K^0 〉. Solve the Schr¨odinger equation to find its state for all time afterwards. Why is it easier to solve this problem using |KU 〉, |KS 〉 vectors rather than |K^0 〉, | K¯^0 〉 vectors? Calculate and plot the probability of finding the meson in state |K^0 〉 as a function of time.

(The neutral K meson system is extraordinarily interesting. I have oversimplified by ignoring decay. More complete treatments can be found in Lipkin, Das & Melissinos, Feynman, and Baym.)

Chapter 6

Continuum Systems

6.1 The states {|p〉} constitute a continuum basis In lecture we showed that the inner product 〈x|p〉 must have the form

〈x|p〉 = C ei(p/¯h)x^ (6.1)

where C may be chosen for convenience.

a. Show that the operator Aˆ =

−∞

dp |p〉〈p| (6.2)

is equal to 2 π¯h|C|^2 ˆ 1 (6.3) by evaluating 〈φ| Aˆ|ψ〉 = 〈φ|ˆ 1 Aˆˆ 1 |ψ〉 (6.4) for arbitrary states |ψ〉 and |φ〉. Hints: Set the first ˆ1 equal to

−∞ dx^ |x〉〈x|, the second ˆ1 equal to

−∞ dx ′ (^) |x′〉〈x′|. The identity

δ(x) =

2 π

−∞

dk eikx^ (6.5)

(see Winter eqn. (3.2-7) p. 107) for the Dirac delta function is useful here. Indeed, this is one of the most useful equations to be found anywhere! b. Using the conventional choice C = 1/

2 π¯h, show that

〈p|p′〉 = δ(p − p′). (6.6)

The expression (6.5) is again helpful.

a. Use the Hermiticity of ˆp to show that

〈p| Hˆ|ψ(t)〉 = p^2 2 m ψ˜(p; t) + 〈p| Vˆ |ψ(t)〉. (6.15)

Now we must investigate 〈p| Vˆ |ψ(t)〉. b. Show that 〈p| Vˆ |ψ(t)〉 =

√^1

2 π¯h

−∞

dx e−i(p/¯h)x^ V (x)ψ(x; t) (6.16)

by inserting the proper form of ˆ1 at the proper location. c. Define the (modified) Fourier transform V˜ (p) of V (x) by

V˜ (p) = √^1 2 π¯h

−∞

dx e−i(p/¯h)xV (x) (6.17)

=

−∞

dx 〈p|x〉V (x). (6.18)

Note that V˜ (p) has funny dimensions. Show that

V (x) =

2 π¯h

−∞

dp ei(p/¯h)x^ V˜ (p) (6.19)

=

−∞

dp 〈x|p〉 V˜ (p). (6.20)

You may use either forms (6.17) and (6.19) in which case the proof employs equation (6.5), or forms (6.18) and (6.20) in which case the proof involves completeness and orthogonality of basis states. d. Hence show that 〈p| Vˆ |ψ(t)〉 =

√^1

2 π¯h

−∞

dp′^ V˜ (p − p′) ψ˜(p′; t). (6.21)

(Caution! Your intermediate expressions will probably involve three distinct variables that you’ll want to call “p”. Put primes on two of them!) e. Put everything together to see that ψ˜(p; t) obeys the integro-differential equation

i¯h ∂ ψ˜(p; t) ∂t

p^2 2 m ψ˜(p; t) + √^1 2 π¯h

−∞

dp′^ V˜ (p − p′) ψ˜(p′; t). (6.22)

This form of the Schr¨odinger equation is particularly useful in the study of superconductivity.

Chapter 7

The Free Particle

7.1 Energy eigenstates In lecture we examined the behavior of a free particle in a state of definite momentum. Such states have a definite energy, but they are not the only possible states of definite energy.

a. Show that the state |ρ(0)〉 = A|p 0 〉 + B| − p 0 〉 (7.1) where |A|^2 + |B|^2 = 1 has definite energy E(p 0 ) = p^20 / 2 m. (That is, |ρ(0)〉 is an energy eigenstate with eigenvalue p^20 / 2 m). b. Show that the “wavefunction” corresponding to |ρ(t)〉 evolves in time as

ρ(x; t) =

√^1

2 π¯h

[ A ei(p^0 x−E(p^0 )t)/¯h^ + B ei(−p^0 x−E(p^0 )t)/¯h]. (7.2)

I use the term wavefunction in quotes because ρ(x; t) is not 〈x|normal state〉 but rather a sum of two terms like 〈x|continuum basis state〉. c. Show that the “probability density” |ρ(x; t)|^2 is independent of time and given by

|ρ(x; t)|^2 =

2 π¯h [1 + 2 Re{A∗B} cos

2 p 0 x ¯h

• 2 Im{A∗B} sin

2 p 0 x ¯h

]. (7.3)