Foundations of Quantum Mechanics

AdminJames Currie

james.currie@durham.ac.uk

Talks every Thursday 13:00 CG91 (chemistry)

Slides available after the lecture at my profile page:

www.ippp.dur.ac.uk -> The Institute -> Research Staff

Obviously no homework or workshops or exam (yay!)

Why more talks on QM?• QM is universal: 117 years old, no deviations found

• QM is (too) useful: applications studied a lot at university, not much time for discussing its meaning (or if it has one)

• QM is radical:

“Anyone who is not shocked by quantum theory has not understood it.” [Bohr]

• QM may be incomplete: many outstanding problems that go beyond calculating things… need a better understanding

• QM is interesting (hopefully to be demonstrated by Easter)

Series Outline1. How to be a Quantum Mechanic

2. Entanglement and decoherence

3. A Gordian knot and Heisenberg’s cut

4. Local hidden realism: Einstein’s “reasonable” solution

5. QM’s classical inheritance

6. Bohmian realism: non-local hidden variables and holism

7. How many cats does it take to solve a paradox?

What I will not be talking about

Foundations of QM is a huge field; many things will not be covered:

• Nelson’s stochastic QM

• Objective reduction models (GRW, Schrödinger-Newton states)

• Consistent histories “Copenhagen done right” (supposedly)

• Epistemic interpretations (QBism) for a good introduction see [arXiv:1311.5253]

This reflects my personal bias and should not stop you looking into these if you’re interested!

Some Excellent BooksThe following are some relevant books I recommend:

• Beyond measure: Jim Baggot. Accessible overview

• The Speakable and Unspeakable in Quantum Mechanics: John Bell. A collection of his papers, all classics

• Foundations and Interpretation of Quantum Mechanics: Gennaro Auletta. Advanced and comprehensive

• Decoherence and the Quantum to Classical Transition: Maximillion Schlosshauer. A modern treatment of QM and decoherence. Also a short introduction article [arXiv:quant-ph/0312059]

• The Quantum Theory of Motion: Peter Holland. The most complete text on de Broglie-Bohm theory

Lecture 1How to be a Quantum Mechanic

• For a point particle, ‘state’ at any given time is given by particle’s position and velocity (or momentum)

• can be formulated as a point in a 6 dimensional space (see lecture 5)

• the classical state evolves deterministically in time according to Newton’s laws of motion

• ‘Observables’ (properties) are functions of the state, e.g. potential energy, V(x)

Classical StatesBefore we think about QM, what is a classical ‘state’?

Quantum StatesA quantum ‘system’ is also specified by a ‘state’:

• system’s degrees of freedom reflected in a ‘Hilbert’ space (a type of vector space). e.g. spin 1/2 system has 2-dimensional Hilbert space

• quantum state is just a point in this space, a vector, denoted by

• quantum states have complex-valued components, e.g.

• ‘inner’ or ‘dot’ product of two quantum states tells us how much they overlap in the Hilbert space, gives a complex number

!

!

Isn’t this all a bit abstract? Yes!

|ψ⟩|φ⟩

⟨φ|ψ⟩

H ⇠ C2

| i✓c1c2

◆with c1, c2 2 C

h�| i ⇠ C

Superpositions and BasesQuantum states are vectors, so we can add them together

!

e.g 2-d example in component form:

!

!

Each Hilbert space is spanned by a set of basis vectors

We can decompose any state into a superposition of orthonormal basis vectors

!

The expansion coefficients tell us how much of is in the basis state

| i = |�i+ |⌘i

✓ 1

2

◆=

✓�1�2

◆+

✓⌘1⌘2

◆=

✓�1 + ⌘1�2 + ⌘2

◆

|ni

| i =X

n

cn|ni

|nicn | i

WavefunctionsIf we are interested in a state with continuous degrees of freedom, like position, then we expand the state in the basis of position basis states :

!

The expansion coefficient in this continuous basis is a ‘wavefunction’

In this basis, the dot product between two states is just an integral:

!

Could also expand in a continuous momentum basis,

!

A momentum wavefunction is just the Fourier transform of a position wavefunction

|xi

| i =Z

dx (x) |xi

h�| i =Z

dx �⇤(x) (x)

(x)

| i =Z

dp (p) |pi

‘Observables’Observables (measurables) in classical mechanics are simple functions of the system’s state, i.e. property values, e.g. spring potential energy

!

Measurables in QM are represented by Hermitian linear operators in the Hilbert space, e.g. for 2-d spin 1/2, spin along z-axis is associated with the operator,

!

Each operator has a special set of values associated with it, called ‘eigen-values’, the closest thing to a classical property value, e.g.

!

When we do a measurement, the only possible outcome is an eigenvalue

V (x) =1

2kx

2

Sz =~2

✓1 00 �1

◆

�± = ±~2

Eigen-statesEigenvalues are definite property values of the state, like in classical mechanics

These are closely associated with a special set of quantum states called “eigenstates”; each eigenvalue corresponds to an eigenstate

!

The action of an observable’s operator on an eigenstate yields an eigenvalue

For any observable, its eigenstates form a complete basis for the Hilbert space, so can expand any quantum state as a superposition of eigenstates

| i =X

n

cn|ni

O|ni = en|ni

e-e link• If a quantum system is in an eigenstate of an observable then we

can assign a value to that state for the observable

• If we do an experiment we always measure an eigenvalue of the observable’s operator and claim the system is in an eigenstate

!

!

This is the eigenvalue-eigenstate (e-e) link

• crucial to make the connection between quantum state and “reality”

What if the system is not in an eigenstate?

system in an eigenstate

measure an eigenvalue

ProbabilityIf the system is not in an eigenstate, then can’t assign a value to it for observables

• must be a superposition of eigenstates

!

• After measurement we claim the system is in an eigenstate

• discontinuous jump (collapse) from a superposition to eigenstate

Can’t say which state it will jump to (indeterminate) but can calculate probability of finding the state in the nth eigenstate upon measurement

!

or for a continuous basis like position:

| i =X

n

cn|ni

P (n) = |cn|2

P (x+ dx) = | (x)|2dx

Expectation valuesIf quantum states jump indeterminately into eigenstates upon measurement:

• can’t predict an outcome for any given measurement

• but can study averages over many measurements

Average value for an observable given by the expectation value for a particular state

!

or in the position basis,

hOi = h |O| i =Z

dx ⇤(x)O(x) (x)

hOi = h |O| i =X

i,j

c⇤i cjhi|O|ji

CompatibilityIn CM all property values can be defined simultaneously. In QM only eigenstates have definite values

Eigenstates are inherent (‘eigen') to the observable’s operator:

• if operators have different eigenstates then system can’t be in an eigenstate of both simultaneously

• properties cannot be determined simultaneously; such observables are incompatible

Mathematically, compatible operators commute

!

Incompatible observables result in an uncertainty relation for measurements

[A, B] = AB � BA = 0

�A�B � 1

2|hCi |, if [A, B] = iC

Time evolutionUntil a measurement, quantum states evolve deterministically

!

If the state is an energy eigenstate,

!

then the time dependence is just a global phase

!

and expectation values are constant in time

H| i = E| i

| (t)i = eiEt/~| (t0)i

h (t)|O| (t)i = h (t0)|eiEt/~Oe�iEt/~| (t0)i = h (t0)|O| (t0)i

i~ @@t

| (t)i = H| (t)i

Example 1Spin 1/2 system:

• Hilbert space 2-d, can write as a column vector. Consider a state,

!

• observable “spin in z direction” represented by operator

!

• this operator has eigenstates with eigenvalues

!

• Probability the state is found in each eigenstate given by

!

Sz =~2

✓1 00 �1

◆

✓ 1

2

◆=

1p10

✓i3

◆

⇢✓10

◆,+

~2

� ⇢✓01

◆,�~

2

�

P (+~/2) = |c1|2 =1

10, P (�~/2) = |c2|2 =

9

10

expectation value for spin of this state

!

spin in x or y direction are incompatible observables with spin along z, e.g.

!

with eigenstates and eigenvalues,

!

!

yielding the uncertainty relations,

!

e.g. for our state,

hSzi =

✓1

10� 9

10

◆~2= �2~

5

Sx

=~2

✓0 11 0

◆, S

y

=~2

✓0 �ii 0

◆

[Sx

, Sy

] = i~ Sz

, [Sz

, Sx

] = i~ Sy

, [Sy

, Sz

] = i~ Sx

⇢1p2

✓11

◆,+

~2

�

S+x

⇢1p2

✓1�1

◆,�~

2

�

S�x

,

⇢1p2

✓1i

◆,+

~2

�

S+y

⇢1p2

✓1�i

◆,�~

2

�

S�y

�Sx

�Sy

� ~2|hS

z

i

|

�Sx

�Sy

� ~25

Example 2Infinite square well

• infinite dimensional Hilbert space

• can expand in continuous position basis… wavefunction

• Energy observable in this basis

!

• energy eigenstates and eigenvalues given by,

!

• can expand any wavefunction in eigenstates

L2[0, L]

(x) = hx| i

⇢ n(x) =

1p2sin

✓n⇡x

L

◆,

n

2~2⇡2

2mL

2

�

H(x) = � ~22m

r2

�(x) =1X

n=1

cn n(x)

consider the superposition at an instant in time

!

probability to be found in nth state,

!

energy expectation value for this state,

!

!

!

!

!

�(x) =1p3 1(x) +

r2

3 3(x)

hEi� = h�|H|�i

=

Zdx �(x)⇤H(x)�(x)

=

Zdx

1

3 1(x)

⇤H(x) 1(x) +

p2

3 1(x)

⇤H(x) 3(x) +

p2

3 3(x)

⇤H(x) 1(x) +2

3 3(x)

⇤H(x) 3(x)

�

=

Zdx

1

3E1| 1(x)|2 +

p2

3E3 1(x)

⇤ 3(x) +

p2

3E1 3(x)

⇤ 1(x) +2

3E3| 3(x)|2

�

=1

3E1 +

2

3E3 =

19~2⇡2

6mL2

|h n|�i|2 =

����Z

dx n(x)�(x)

����2

=

����X

i

ci

Zdx n(x) i(x)

����2

= |cn|2

Time dependence of wavefunction from TDSE

!

solve differential equation with usual methods

!

!

!

the state will evolve in time, e.g.

!

probability it is found in initial state again after a time t,

i~ @

@t (x, t) = H (x, t)

�(x, t) = (x)F (t)

,! (x, t) =1X

n=1

e

�iEnt/~cn n(x)

�(x, t) =1p3e

�iE1t/~ 1(x) +

r2

3e

�iE3t/~ 3(x)

|h�(0)|�(t)i|2 =

����1

3

e�i�Et/~+

2

3

e+i�Et/~����2

=

5

9

+

4

9

cos

✓8~⇡2t

mL2

◆

Summary• QM systems’ degrees of freedom are encoded in a Hilbert space

• QM states are vectors in this space and add linearly

• observables are linear operators in this space, they do not generally commute

• each operator has a special set of numbers (eigenvalues), values of the observable

• the outcome of a measurement is always an eigenvalue

• eigenvalues are associated with eigenstates via the e-e- link

• incompatible observables cannot share eigenstates, implies uncertainty relations

• we can always decompose any state into a superposition of eigenstates

• measurements are indeterminate but we can calculate averages and probabilities

• the state evolves according to the linear Schrödinger equation until a measurement

Series Outline1. How to be a Quantum Mechanic

2. Entanglement and decoherence

3. A Gordian knot and Heisenberg’s cut

4. Local hidden realism: Einstein’s “reasonable” solution

5. QM’s classical inheritance

6. Bohmian realism: non-local hidden variables and holism

7. How many cats does it take to solve a paradox?