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Quantum Mechanics 8.

Lecture notes from Prof. B. Zwiebach

Edited by

Ing. Gioacchino Salemme

August 1, 2018

Contents

• 1 Lecture 1: Introduction to Quantum Mechanics
  • 1.1 Linearity of the equations of motion
  • 1.2 Complex Numbers are Essential
  • 1.3 Loss of Determinism
  • 1.4 Quantum Superpositions
  • 1.5 Entanglement
• 2 Lecture 2: Experiments with photons
  • 2.1 Mach-Zehder Interferometer
  • 2.2 Elitzur-Vaidman Bombs
• 3 Lecture 3: Experiments with photons
  • 3.1 Photoelectric Effect
  • 3.2 Compton Scattering
  • 3.3 Matter waves
• 4 Lecture 4: The De Broglie Equation
  • 4.1 De Broglie wavelength and Galilean transformations
  • 4.2 Phase and Group Velocities
  • 4.3 Choosing the wavefunction for a free particle
• 5 Lecture 5: The wavefunction and the Schrödinger equation
  • 5.1 Equations for a wavefunction
  • 5.2 Schrödinger equation for particle in a potential
  • 5.3 Interpreting the Wavefunction
• 6 Lecture
  • 6.1 Normalization and time evolution
  • 6.2 The wave function as probability amplitude
  • 6.3 The probability current
  • 6.4 Probability current in 3D and current conservation
• 7 Lecture
  • 7.1 Wave packets and Uncertainty
  • 7.2 Wavepacket Shape Changes
  • 7.3 Time evolution of a free wave packet
• 8 Lecture 8: Momentum space and expectations values of operators
  • 8.1 Uncovering momentum space
  • 8.2 Expectation Values of Operators
  • 8.3 Time dependence of expectation values
• CONTENTS
• 9 Lecture
  • 9.1 Observables and Hermitian operators
  • 9.2 Uncertainty
• 10 Lecture 10: Solving the Time-Independent Schrödinger Equation
  • 10.1 Stationary States
  • 10.2 Solving for Energy Eigenstates
  • 10.3 Free particle on a circle.
• 11 Lecture 11: Infinite and finite square well
  • 11.1 The Infinite Square Well
  • 11.2 The Finite Square Well
• 12 Lecture 12: Property of solutions to the Schrödinger equation in 1-D
  • 12.1 General properties
  • 12.2 Bound states in slowly varying potentials
  • 12.3 Sketching wavefunction behavior in different regions
  • 12.4 Shooting Method
• 13 Lecture 13: Delta Function Potential and Simple Harmonic Oscillator
  • 13.1 The Delta Function Potential
  • 13.2 The Node Theorem
  • 13.3 Harmonic Oscillator
• 14 Lecture 14 and 15: Algebraic approach to the SHO
  • 14.1 Algebraic Solution of the Oscillator
  • 14.2 Operator manipulation and the spectrum
• 15 Lecture 16: Scattering States and the Step Potential
  • 15.1 The Step Potential
  • 15.2 Step Potential with E > V
  • 15.3 Step Potential with E < V
  • 15.4 Wavepackets in the step potential
• 16 Lecture 17: Resonant transmission and Ramsauer-Townsend
  • 16.1 Resonant transmission in a square well
  • 16.2 The Ramsauer-Townsend Effect
• 17 Lecture 18: Scattering in one dimension
  • 17.1 Scattering in One Dimension
  • 17.2 Time Delay
  • 17.3 An Example
• 18 Lecture 19:
  • 18.1 Levinson’s Theorem
  • 18.2 Resonances
• 19 Lecture 20: Quantum Mechanics in 3D and Central Potentials
  • 19.1 Schrödinger equation in 3D and Angular Momentum
  • 19.2 The angular momentum operator
  • 19.3 Eigenstates of Angular Momentum
  • 19.4 The Radial Wave Equation
• CONTENTS
• 20 Lecture 21: Hydrogen Atom
  • 20.1 The Hydrogen Atom
  • 20.2 Hydrogen atom spectrum
• 21 Lecture 23: Hydrogen Atom
  • 21.1
• A Trasformate di Fourier
  • A.1 Definizione di Trasformata di fourier
    • A.1.1 Trasformata di Fourier di una porta
    • A.1.2 Trasformata di Fourier della delta di Dirac
    • A.1.3 Antitrasformata di Fourier

Chapter 1

Lecture 1: Introduction to Quantum

Mechanics

Contents

1.1 Linearity of the equations of motion.................... 6

1.2 Complex Numbers are Essential...................... 8

1.3 Loss of Determinism.............................. 9

1.4 Quantum Superpositions........................... 11

1.5 Entanglement.................................. 16

Quantum mechanics is now almost one-hundred years old, but we are still discovering some of its surprising features and it remains the subject of much investigation and speculation. The framework of quantum mechanics is a rich and elegant extension of the framework of classical physics. It is also counterintuitive and almost paradoxical. Quantum physics has replaced classical physics as the correct fundamental description of our physical universe. It is used routinely to describe most phenomena that occur at short distances. Quantum physics is the result of applying the framework of quantum mechanics to different physical phenomena. We thus have Quantum Electrodynamics, when quantum mechanics is applied to electromagnetism, Quantum Optics, when it is applied to light and optical devices, or Quantum Gravity, when it is applied to gravitation. Quantum mechanics indeed provides a remarkably coherent and elegant framework. The era of quantum physics begins in 1925, with the discoveries of Schrödinger and Heisenberg. The seeds for these discoveries were planted by Planck, Einstein, Bohr, de Broglie, and others. It is a tribute to human imagination that we have been able to discover the counterintuitive and abstract set of rules that define quantum mechanics. Here we aim to explain and provide some perspective on the main features of this framework. We will begin by discussing the property of linearity, which quantum mechanics shares with electromagnetic theory. This property tells us what kind of theory quantum mechanics is and why, it could be argued, it is simpler than classical mechanics. We then turn to photons, the particles of light. We use photons and polarizers to explain why quantum physics is not deterministic and, in contrast with classical physics, the results of some experiments cannot be predicted. Quantum mechanics is a framework in which we can only predict the probabilities for the various outcomes of any given experiment. Our next subject is quantum superpositions, in which a quantum object somehow manages to exist simultaneously in two mutually incompatible states. A quantum light-bulb, for example, could be in a state in which it is both on and off at the same time!

1.1 Linearity of the equations of motion 7

where a is a number. Note that these conditions imply that

L(αu 1 + βu 2 ) = αLu 1 + βLu 2 (1.1.5)

showing that if u 1 is a solution (Lu 1 = 0) and u 2 is a solution (Lu 2 = 0) then αu 1 + βu 2 is also a solution. We call αu 1 + βu 2 the general superposition of the solutions u 1 and u 2. An example may help. Consider the equation du dt

τ

u = 0 (1.1.6)

where τ is a constant with units of time. This is, in fact, a linear differential equation, and takes the form Lu = 0 if we define Lu =

du dt

τ

u (1.1.7)

Einstein’s theory of general relativity is a nonlinear theory whose dynamical variable is a gravitational field, the field that describes, for example, how planets move around a star. Being a nonlinear theory, you simply cannot add the gravitational fields of different solutions to find a new solution. This makes Einstein’s theory rather complicated, by all accounts much more complicated than Maxwell’s theory. In fact, classical mechanics, as invented mostly by Isaac Newton, is also a nonlinear theory! In classical mechanics the dynamical variables are positions and velocities of particles, acted by forces. There is no general way to use two solutions to build a third. Indeed, consider the equation of motion for a particle on a line under the influence of a time-independent potential V (x), which is in general an arbitrary function of x. The dynamical variable in this problem is x(t), the position as a function of time. Letting V ′^ denote the derivative of V with respect to its argument, Newton’s second law takes the form

m

d^2 x(t) dx^2

= −V ′(x) (1.1.8)

The left-hand side is the mass times acceleration and the right hand side is the force experienced by the particle in the potential. It is probably worth to emphasize that the right hand side is the function V ′(x) evaluated for x set equal to x(t):

V ′(x(t)) =

∂V (x) ∂x

x=x(t)

While we could have used here an ordinary derivative, we wrote a partial derivative as is commonly done for the general case of time dependent potentials. The reason equation (1.1.8) is not a linear equation is that the function V ′(x) is not linear. In general, for arbitrary functions u and v we expect V ′(au) 6 = aV ′(u), and V ′(u + v) 6 = V ′(u) + V ′(v) (1.1.10)

As a result, given a solution x(t), the scaled solution αx(t) is not expected to be a solution. Given two solutions x 1 (t) and x 2 (t) then x 1 (t) + x 2 (t) is not guaranteed to be a solution either. Quantum mechanics is a linear theory. The signature equation in this theory, the so-called Schrödinger equation is a linear equation for a quantity called the wavefunction and it determines its time evolution. The wavefunction is the dynamical variable in quantum mechanics but, curiously, its physical interpretation was not clear to Erwin Schrödinger when he wrote the equation in

1925. It was Max Born, who months later suggested that the wavefunction encodes probabilities. This was the correct physical interpretation, but it was thoroughly disliked by many, including Schrödinger, who remained unhappy about it for the rest of his life. The linearity of quantum mechanics implies a profound simplicity. In some sense quantum mechanics is simpler than classical mechanics. In quantum mechanics solutions can be added to form new solutions.

Lecture 1: Introduction to Quantum Mechanics 8

The wavefunction Ψ depends on time and may also depend on space. The Schrödinger equation is a partial differential equation that takes the form

iℏ

∂t

= HˆΨ (1.1.11)

where the Hamiltonian (or energy operator) Hˆ is a linear operator that can act on wavefunctions:

H^ ˆ(aΨ) = a HˆΨ, Hˆ(Ψ 1 + Ψ 2 ) = HˆΨ 1 + HˆΨ 2 (1.1.12)

with a constant that in fact need not be real; it can be a complex number. Of course, Hˆ itself does not depend on the wavefunction! To check that the Schrödinger equation is linear we cast it in the form LΨ = 0 with L defined as

LΨ = iℏ

∂t

− HˆΨ (1.1.13)

It is now a simple matter to verify that L is a linear operator. Physically this means that if Ψ 1 and Ψ 2 are solutions to the Schrödinger equation , then so is the superposition αΨ 1 + βΨ 2 , where α and β are both complex numbers, i.e. (α, β ∈ C)

1.2 Complex Numbers are Essential

Quantum mechanics is the first physics theory that truly makes use of complex numbers. The numbers most of us use for daily life (integers, fractions, decimals) are real numbers. The set of complex numbers is denoted by C and the set of real numbers is denoted by R. Complex numbers appear when we combine real numbers with the imaginary unit i, defined to be equal to the square root of minus one: i =

− 1. Being the square root of minus one, it means that i squared must give minus one: i^2 = − 1. Complex numbers are fundamental in mathematics. An equation like x^2 = − 4 , for an unknown x cannot be solved if x has to be real. No real number squared gives you minus 4. But if we allow for complex numbers, we have the solutions x = ± 2 i. Mathematicians have shown that all polynomial equations can be solved in terms of complex numbers. A complex number z, in all generality, is a number of the form

z = a + ib, a, b ∈ R (1.2.1)

Here a and b are real numbers, and ib denotes the product of i with b. The number a is called the real part of z and b is called the imaginary part of z:

Re(z) = a, Im(z) = b (1.2.2)

The complex conjugate z∗^ of z is defined by

z∗^ = a − ib (1.2.3)

You can quickly verify that a complex number z is real if z∗^ = z and it is purely imaginary if z∗^ = −z. For any complex number z = a + ib, one can define the norm |z| of the complex number to be a positive, real number given by

|z| =

a^2 + b^2 (1.2.4)

You can quickly check that z^2 = z · z∗^ (1.2.5)

Lecture 1: Introduction to Quantum Mechanics 10

photons, are routinely manipulated in laboratories around the world. Even if mysterious, we have grown accustomed to them. Each photon of visible light carries very little energy a small laser pulse can contain many billions of photons. Our eye, however, is a very good photon detector: in total darkness, we are able to see light when as little as ten photons hit upon our retina. When we say that light behaves like a particle we mean a quantum mechanical particle: a packet of energy and momentum that is not composed of smaller packets. We do not mean a classical point particle or Newtonian corpuscle, which is a zero-size object with definite position and velocity. As it turns out, the energy of a photon depends only on the color of the light. As Einstein discovered, the energy E and frequency ν for a photon are related by

E = hν (1.3.1)

where h is the Plank constant, whose value is

h = 4. 136 × 10 −^15 eV s (1.3.2)

The frequency of a photon determines the wavelength λ of the light through the relation νλ = c, where c is the speed of light. All green photons, for example, have the same energy. To increase the energy in a light beam while keeping the same color, one simply needs more photons. As we now explain, the existence of photons implies that Quantum Mechanics is not deterministic. By this we mean that the result of an experiment cannot be determined, as it would in classical physics, by the conditions that are under the control of the experimenter. Consider a polarizer whose preferential direction is aligned along the x direction, as shown in Figure 1. Light that is linearly polarized along the x direction namely, light whose electric field points in this direction, goes through the polarizer. If the incident light polarization is orthogonal to the x direction the light will not go through at all. Thus light linearly polarized in the y direction will be totally absorbed by the polarizer. Now consider light polarized along a direction forming an angle α with the x-axis, as shown in Figure 2. What happens? Thinking of the light as a propagating wave, the incident electric field Eα makes an angle α with the x-axis and therefore takes the form

Eα = E 0 cos αxˆ + E 0 sin αˆy (1.3.3)

This is an electric field of magnitude E 0. In here we are ignoring the time and space dependence of the wave; they are not relevant to our discussion. When this electric field hits the polarizer, the component along xˆ goes through and the component along yˆ is absorbed. Thus

Beyond the polarizer: E = E 0 cos α xˆ (1.3.4)

You probably recall that the energy in an electromagnetic wave is proportional to the square of the magnitude of the electric field. This means that the fraction of the beam’s energy that goes through the polarizer is (cos α)^2. It is also well known that the light emerging from the polarizer has the same frequency as the incident light. So far so good. But now, let us try to understand this result by thinking about the photons that make up the incident light. The premise here is that all photons in the incident beam are identical. Moreover the photons do not interact with each other. We could even imagine sending the whole energy of the incident light beam one photon at a time. Since all the light that emerges from the polarizer has the same frequency as the incident light, and thus the same frequency, we must conclude that each individual photon either goes through or is absorbed. If a fraction of a photon went through it would be a photon of lower energy and thus lower frequency, which is something that does not happen.

1.4 Quantum Superpositions 11

But now we have a problem. As we know from the wave analysis, roughly a fraction (cos α)^2 of the photons must go through, since that is the fraction of the energy that is transmitted. Consequently a fraction 1(cos α)^2 of the photons must be absorbed. But if all the photons are identical, why is it that what happens to one photon does not happen to all of them? The answer in quantum mechanics is that there is indeed a loss of determinism. No one can predict if a photon will go through or will get absorbed. The best anyone can do is to predict probabilities. In this case there would be a probability (cos α)^2 of going through and a probability 1 - (cos α)^2 of failing to go through. Two escape routes suggest themselves. Perhaps the polarizer is not really a homogeneous object and depending exactly on where the photon his it either gets absorbed or goes through. Experiments show this is not the case. A more intriguing possibility was suggested by Einstein and others. A possible way out, they claimed, was the existence of hidden variables. The photons, while apparently identical, would have other hidden properties, not currently understood, that would determine with certainty which photon goes through and which photon gets absorbed.

Hidden variable theories would seem to be untestable, but surprisingly they can be tested. Through the work of John Bell and others, physicists have devised clever experiments that rule out most versions of hidden variable theories. No one has figured out how to restore determinism to quantum mechanics. It seems to be an impossible task. When we try to describe photons quantum mechanically we could use wavefunctions, or equivalently the language of states. A photon polarized along the ˆx direction is not represented using an electric field, but rather we just give a name for its state : |photon; x〉 (1.3.5)

We will learn the rules needed to manipulate such objects, but for the time being you could think of it like a vector in some space yet to be defined. Another state of a photon, or vector is

|photon; y〉 (1.3.6) representing a photon polarized along ˆy. These states are the wavefunctions that represent the photon. We now claim that the photons in the beam that is polarized along the direction α are in a state |photon; α〉 that can be written as a superposition of the above two states: |photon; α〉 = cos α |photon; x〉 + sin α |photon; y〉 (1.3.7)

This equation should be compared with (1.3.3). While there are some similarities-both are superpositions- one refers to electric fields and the other to states of a single photon. Any photon that emerges from the polarizer will necessarily be polarized in the ˆx direction and therefore it will be in the state Beyond the polarizer: |photon; x〉 (1.3.8)

This can be compared with (1.3.4) which with the factor cos α carries information about the amplitude of the wave. Here, for a single photon, there is no room for such a factor. In the famous Fifth Solvay International Conference of 1927 the worlds most notable physicists gathered to discuss the newly formulated quantum theory. Seventeen out of the twenty nine attendees were or became Nobel Prize winners. Einstein, unhappy with the uncertainty in quantum mechanics stated the nowadays famous quote: "God does not play dice", to which Niels Bohr is said to have answered: "Einstein, stop telling God what to do." Bohr was willing to accept the loss of determinism, Einstein was not.

1.4 Quantum Superpositions

We have already discussed the concept of linearity; the idea that the sum of two solutions representing physical realities represents a new, allowed, physical reality. This superposition of

1.4 Quantum Superpositions 13

BS

BS

M

M

Incident

beam

a

b

Figure 1.2: A Mach-Zehnder interferometer consists of two beam splitters BS1 and BS2, two mirrors M1 and M2, and two detectors D0 and D1. An incident beam will be split into two beams by BS1. One beam goes through the upper branch, which contains M1, the other beam goes through the lower branch, which contains M2. The beams on the two branches recombine at BS2 and are then sent into the detectors. The configuration is prepared to produce an interference so that all incident photons end at the detector D0, with none at D1.

or lower branch. Thus the state of the photon in the interferometer is a funny state in which the photon seems to be doing two incompatible things at the same time. Equation (1.3.7) is another example of a quantum superposition. The photon state has a component along an x-polarized photon and a component along a y-polarized photon. When we speak of a wavefunction, we also sometimes call it a state, because the wavefunction specifies the state of our quantum system. We also sometimes refer to states as vectors. A quantum state may not be a vector like the familiar vectors in three-dimensional space but it is a vector nonetheless because it makes sense to add states and to multiply states by numbers. Just like vectors can be added, linearity guarantees that adding wavefunctions or states is a sensible thing to do. Just like any vector can be written as a sum of other vectors in many different ways, we will do the same with our states. By writing our physical state as sums of other states we can learn about the properties of our state. Consider now two states |A〉 and |B〉. Assume, in addition, that when measuring some property Q in the state |B〉 the answer is always a, and when measuring the same property Q in the state |B〉 the answer is always b. Suppose now that our physical state |Ψ〉 is the superposition

|Ψ〉 = α |A〉 + β |B〉 , α, β ∈ C (1.4.1)

What happens now if we measure property Q in the system described by the state |Ψ〉? It may seem reasonable that one gets some intermediate value between a and b, but this is not what happens. A measurement of Q will yield either a or b. There is no certain answer, classical determinism is lost, but the answer is always one of these two values and not an intermediate one. The coefficients α and β in the above superposition affect the probabilities with which we may obtain the two possible values. In fact, the probabilities to obtain a or b

Probability(a) ∝ |α|^2 , Probability(b) ∝ |β|^2 (1.4.2)

Lecture 1: Introduction to Quantum Mechanics 14

Since the only two possibilities are to measure a or b, the actual probabilities must sum to one and therefore they are given by

Probability(a) =

|α|^2 |α|^2 + |β|^2

If we obtain the value a, immediate repeated measurements would still give a, so the state after the measurement must be |A〉. The same happens for b, so we have After measuring a, the state becomes |Ψ〉 = |A〉 , After measuring b, the state becomes |Ψ〉 = |B〉 In quantum mechanics one makes the following assumption: Superposing a state with itself doesn’t chance the physics, nor does it change the state in a non-trivial way. Since superimposing a state with itself simply changes the overall number multiplying it, we have that Φ and αΨ represent the same physics for any complex number α different from zero. Thus, letting u represent physical equivalence |A〉 u 2 |A〉 u i |A〉 − u |A〉 (1.4.4) This assumption is necessary to verify that the polarization of a photon state has the expected number of degrees of freedom. The polarization of a plane wave, as one studies in electromagnetism, is described by two real numbers. For this consider an elliptically polarized wave, as shown in Figure 1.3. At any given point, the electric field vector traces an ellipse whose shape is encoded by the ratio a/b of the semi-major axes (the first real parameter) and a tilt encoded by the angle θ (the second real parameter).

x

y

a b

θ

Figure 1.3: Parameters that define an elliptically polarized state.

Consider for this a general photon state formed by superposition of the two independent polarization states |photon; x〉 and |photon; y〉 α |photon; x〉 + β |photon; y〉 , α, β ∈ C (1.4.5)

Lecture 1: Introduction to Quantum Mechanics 16

z

Spin up

z

Spin down

Figure 1.4: An electron with spin along the z axis. Left: the electron is said to have spin up along z. Right: the electron is said to have spin down along z. The up and down arrows represent the direction of the angular momentum associated with the spinning electron.

of |Ψ〉 states, however, we would find a very simple result: all states pointing up along x. The critic’s ensemble is not equivalent to our quantum mechanical ensemble. The critic is thus shown wrong in his or her attempt to show that quantum mechanical superpositions are not required.

1.5 Entanglement

When we consider superposition of states of two particles we can get the remarkable phenomenon called quantum mechanical entanglement. Entangled states of two particles are those in which we cant speak separately of the state of each particle. The particles are bound together in a common state in which they are entangled with each other. Let us consider two non-interacting particles. Particle 1 could be in any of the states

{|u 1 ,〉 , |u 2 〉 ,... } (1.5.1)

while particle 2 could b in any of the states

{|v 1 ,〉 , |v 2 〉 ,... } (1.5.2)

It may seem reasonable to conclude that the state of the full system, including particle 1 and particle 2 would be specified by stating the state of particle 1 and the state of particle 2. If that would be the case the possible states would be written as

|ui〉 ⊗ |vj 〉 , i, j ∈ N (1.5.3)

for some specific choice of i and j that specify the state of particle one and particle two, respectively. Here we have used the symbol ⊗, which means tensor product, to combine the two states into a single state for the whole system. We will study ⊗ later, but for the time being we can think of it as a kind of product that distributes over addition and obeys simple rules, as follows

(α 1 |u 1 〉 + α 2 |u 2 〉) ⊗ (β 1 |v 1 〉 + β 2 |v 2 〉) = α 1 β 1 |u 1 〉 ⊗ |v 1 〉 + α 1 β 2 |u 1 〉 ⊗ |v 2 〉

• α 2 β 1 |u 2 〉 ⊗ |v 1 〉 + α 2 β 2 |u 2 〉 ⊗ |v 2 〉

The numbers can be moved across the ⊗ but the order of the states must be preserved. The state on the left-hand side expanded out on the right-hand side is still of the type where of the first particle (α 1 |u 1 〉 + α 2 |u 2 〉) with a state of the second particle (β 1 |v 1 〉 + β 2 |v 2 〉). Just like any one of the states listed in (1.5.3) this state is not entangled.

1.5 Entanglement 17

Using the states in (1.5.3), however, we can construct more intriguing superpositions. Consider the following one |u 1 〉 ⊗ |v 1 〉 + |u 2 〉 ⊗ |v 2 〉 (1.5.5)

A state of two particles is said to be entangled if it cannot be written in the factorized form (... ) ⊗ (... ) which allows us to describe the state by simply stating the state of each particle. We can easily see that the state (1.5.5) cannot be factorized. If it could it would have to be with a product as indicated in (1.5.5). Clearly, involving states like |u 3 〉 or |v 3 〉 that do not appear in (1.5.5) would not help. To determine the constants α 1 , α 2 , β 1 , β 2 we compare the right hand side of (1.5.4) with our state and conclude that we need

α 1 β 1 = 0, α 1 β 2 = 0, α 2 β 1 = 0, α 2 β 2 = 0 (1.5.6)

It is clear that there is no solution here. The second equation, for example, requires either α 1 or β 2 to be zero. Having α 1 = 0 contradicts the first equation, and having β 2 = 0 contradicts the last equation. This confirms that the state (1.5.5) is indeed an entangled state. There is no way to describe the state by specifying a state for each of the particles. Let us illustrate the above discussion using electrons and their spin states. Consider a state of two electrons denoted as |↑〉 ⊗ |↓〉. As the notation indicates, the first electron, described by the first arrow, is up along z while the second electron, described by the second arrow, is down along z (we omit the label z on the state for brevity). This is not an entangled state. Another possible state is one where they are doing exactly the opposite: in |↓〉 ⊗ |↑〉 the first electron is down and the second is up. This second state is also not entangled. It now follows that by superposition we can consider the state

This is a entangled state of the pair of electrons. In the state (1.5.7) the first electron is up along z if the second electron is down along z (first term), or the first electron is down along z if the second electron is up along z (second term). There is a correlation between the spins of the two particles; they always point in opposite directions. Imagine that the two entangled electrons are very far away from each other: Alice has one electron of the pair on planet earth and Bob has the other electron on the moon. Nothing we know is connecting these particles but nevertheless the states of the electrons are linked. Measurements we do on the separate particles exhibit correlations. Suppose Alice measures the spin of the electron on earth. If she finds it up along z, it means that the first summand in the above superposition is realized, because in that summand the first particle is up. As discussed before, the state of the two particles immediately becomes that of the first summand. This means that the electron on the moon will instantaneously go into the spin down-along-z configuration, something that could be confirmed by Bob, who is sitting in the moon with that particle in his lab. This effect on Bob’s electron happens before a message, carried with the speed of light, could reach the moon telling him that a measurement has been done by Alice on the earth particle and the result was spin up. Of course, experiments must be done with an ensemble that contains many pairs of particles, each pair in the same entangled state above. Half of the times the electron on earth will be found up, with the electron on the moon down and the other half of the times the electron on earth will be found down, with the electron on the moon up. Our friendly critic could now say, correctly, that such correlations between the measurements of spins along z could have been produced by preparing a conventional ensemble in which 50% of the pairs are in the state |↑〉 ⊗ |↓〉 and the other 50% of the pairs are in the state |↓〉 ⊗ |↑〉. Such objections were dealt with conclusively in 1964 by John Bell, who showed that if Alice and Bob are able to measure spin in three arbitrary directions, the correlations predicted by the quantum entangled state are different from the classical correlations of any conceivable

Chapter 2

Lecture 2: Experiments with photons

Contents

2.1 Mach-Zehder Interferometer......................... 19 2.2 Elitzur-Vaidman Bombs........................... 24

2.1 Mach-Zehder Interferometer

We have discussed before the Mach-Zehnder interferometer, which we show again in Figure 2.1. It contains two beam-splitters BS1 and BS2 and two mirrors. Inside the interferometer we have two beams, one going over the upper branch and one going over the lower branch. This extends beyond BS2: the upper branch continues to D0 while the lower branch continues to D1.

BS

BS

M

M

Incident beam

a

b

D

D

phase shifter

Figure 2.1: The Mach-Zehnder Interferometer.

Vertical cuts in the above figure intersect the two beams and we can ask what is the probability to find a photon in each of the two beams at that cut. For this we need two probability amplitudes, or two complex numbers, whose norm-squared would give probabilities. We can encode this information in a two component vector as ( α β

Lecture 2: Experiments with photons 20

Here α is the probability amplitude to be in the upper beam and β the probability amplitude to be in the lower beam. Therefore, |α|^2 would be the probability to find the photon in the upper beam and |β|^2 the probability to find the photon in the lower beam. Since the photon must be found in either one of the beams we must have

|α|^2 + |β|^2 = 1 (2.1.2)

Following this notation, we would have for the cases when the photon is definitely in one or the other beam:

photon on upper beam:

, photon on bottom beam:

We can view the state (2.1.1) as a superposition of these two simpler states using the rules of vector addition and multiplication: ( α β

α 0

β

= α

• β

In the interferometer shown in Figure 2.1 we included in the lower branch a phase shifter, a piece of material whose only effect is to multiply the probability amplitude by a fixed phase eiδ^ with δ ∈ R: the probability amplitude α to the left of the device becomes eiα^ to the right of the device. Since the norm of a phase is one, the phase-shifter does not change the probability to find the photon. When the phase δ is equal to π, the effect of the phase shifter is to change the sign of the wavefunction since eiπ^ = − 1.

Let us now consider the effect of beam splitters in detail. If the incoming photon hits a beam splitter from the top, we consider this photon to belong to the upper branch and represent it by( 1 0

. If the incoming photon hits the beam-splitter from the bottom, we consider this photon to

belong to the lower branch, and represent it by

. We show the two cases in Figure 2.2. The effect of the beam 1 splitter is to give an output wavefunction for each of the two cases:

left BS:

s t

, Right BS:

u v

As you can see from the diagram, for the photon hitting from above, s may be thought as a reflection amplitude and t as a transmission coefficient. Similarly, for the photon hitting from below, v may be thought as a reflection amplitude and u as a transmission coefficient. The four numbers s, t, u, v, by linearity, characterize completely the beam splitter. They can be used to predict the output given any incident photon, which may have amplitudes to hit both from above and from below. Indeed, an incident photon state

α β

would give

( α β

= α

• β

→ α

s t

• β

u v

αs + βu αt + βv

s u t v

α β

In summary, we see that the BS produces the following effect ( α β

s u t v

α β

We can represent the action of the beam splitter as matrix multiplication on the incoming wavefunction, with the two-by-two matrix