1

Advanced Quantum Mechanics

University of York

Lecturer: Rex Godby

Notes by Victor Naden Robinson

Lecture 1: TDSE

Lecture 2: TDSE

Lecture 3: FMG

Lecture 4: FMG

Lecture 5: Ehrenfest’s Theorem and the Classical Limit

Lecture 6: Classical Relations

Lecture 7: Wave broadening and Many Particle Systems

Lecture 8: Identical Particles

Lecture 9: Identical Particles Continued

Lecture 10: Hartree Theory

Lecture 11: Hartree-Fock Theory

Lecture 12: Density Functional Theory

Lecture 13: Density Functional Theory Continued

Lecture 14: Annihilation and Commutator Relations

Lecture 15: Field Operators

Lecture 16: Heisenberg Picture

Lecture 17: Time-dependence of and operators inducing field operators

Lecture 18: Many Body Perturbation Theory and Quantisation of Fields

2

Lecture 1: The Time Dependent Schrödinger Equation

Introduction – this course hopes to explain the TDSE, the classical limit of quantum

mechanics (QM), many particle systems, and second quantisation, over 18 lectures.

The TDSE:

(1.1)

Note is typically used to refer to many particles systems, rather than

( ) ( ) ( )

(1.2)

This is true for any wave function, i.e. not necessarily just Eigen functions of .

Take the cases where

(a) is independent of and ( ) is an Eigen function of

The solution is given by

( ) ( )

(1.3)

Where at time ( )

Such wave functions are often called “stationary” states, because the physical observables are

stationary. For example

∫

∫ ( ) ( )

(1.4)

The exponential components combine to make unity.

(a) is independent of and ( ) is not necessarily an Eigen function of

3

The TDSE is a linear differential equation:

So linear combinations of solutions are themselves solutions.

Completeness properties means that our , at , can always be expressed as a linear

combinations of the Eigen functions of .

( ) ∑

( )

(1.5)

The solution is:

( ) ∑

( )

(1.6)

So for each Eigen function that contributes to the wave function ( ( )) oscillates at its

own rate.

Taking the example of the harmonic oscillator:

( )

√ ( )

( )

√

( )

√

( )

( )

√

{ ( ) ( )}

(1.8)

Examining the { ( ) ( )} part of eqn (1.8) implies

4

Showing the linear combination of solutions

Lecture 2: TDSE

c) Case where is time dependent

Analytical progress is only possible where the time dependence of ( ) is weak

This leads to Time-Dependent perturbation theory.

i.e. write

( ) ( ) ( )

(2.1)

Where the 2nd

term is assumed to be small

Write

( ) ∑

( )

(2.2)

Where … can be replaced with (

)

And

5

In the expectation that for weak ( ), the functions ( ) will be slowly varying (c.f. case

(b)) TDSE

Using TDSE,

∑{ ( )

(

)

}

∑{

( ) ( )}

(2.3)

Where two terms clearly cancel

Now by taking ∫

∑ ( )

∫ ∑

∫ ( ) ( ) ( )

(2.4)

This is only non-zero when

( ) ( ) ∑ ( )

(2.5)

( )

∑ ( )

( )

(2.6)

There are the equations of motion of

Equation (2.6) is solved iteratively, starting from the initial values of ( ) ( )

Let us consider the case where

Thus

( ) ( )

6

(2.7)

( )

∫

( )

( )

(2.8)

Write

( )

And so

( )

∫

( )

(2.9)

This (2.9) is 1st order time-dependent perturbation theory; valid provided has not increased

too much from zero, i.e. has not declined too much from 1.

Sinusoidal time-dependence – Fermi’s Golden Rule

Applies when

( ) {

( ) ( )

Lecture 3: Fermi’s Golden Rule

Following on from Sinusoidal time-dependence – e.g. an EM wave applied to atom

Write

( )

( )

(3.1)

Now using the result from lecture 2 (equation (2.9))

We come to

( )

∫

( ( )

( )

)

(3.2)

7

After integration

( )

[

( )

( )

( )

( )]

(3.3)

( )

( ( )

( )

( )

( ))

(3.4)

This is non-negligible only when is small

[Diagram showing two energy levels and – there energy difference or the

inverse of this]

(3.5)

In equation (3.5) the negative is for emission and positive for absorption.

Neglect the negligible term before taking | |

To recap, 1st order time-dependent perturbation theory is being used to describe EM radiation

interacting with an atom (probably the electron energy levels).

| ( )|

| |

| ( ) |

( )

(3.6)

Note using the trick of switching

Now

| | ( )

| |

| |

8

| | ( )

(3.7)

( )

( )

| |

(3.8)

Returning to (3.5) and using the trick

| ( )|

| |

(( )

)

( )

(3.9)

And the 4’s will cancel. Consider

(( )

)

( )

[Two graphs; the first describes at fixed t, the second describes t at fixed ]

[First is a spectrum (like young’s double slit / Gaussian), second is small constant amplitude]

As long as is not very small, we can approximate this by a delta function:

( )

Where = area under curve,

∫ (( ))

( )

(3.10*)

{Check why the time term leaves the sin argument – looks like mistake in (3.9) or (3.10)}

(( )

) ( )

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Lecture 4: Continuing to FMG

Consider

∫ (( ))

( )

(4.1)

Write:

( )

∫

( )

∫

( )

(4.2)

Thus

| ( )|

| |

( )

(4.3)

Including the other case as well ( ), we can write

| ( )|

| |

{ ( ) ( )}

(4.4)

This is Fermi’s Golden Rule

Note, in the case of static perturbation theory the curly brackets {…} becomes ( )

because we must double before taking the modulus squared.

only has a meaning when we have a distribution of states

[Diagram of two energy levels, K and S, separated as before, K has a distribution of states]

( )

The total probability of ending up in one of these fine states is

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∫ | | ( )

Using (4.3)

∫ | | ( )

| |

( ) ∫ ( )

(4.5)

∫ | | ( )

| |

( )

(4.6)

Note:

1) Probability of being in state . I.e. the transmission rate is constant.

2) Result is valid only if t is sufficiently small so that the probability of not being in a

state is small.

3) Probability if ∫

Matrix Elements and Selection Rules

The maths becomes challenging at times from now on, selection rules will be familiar to those whom

have studied atomic physics.

To start, consider the example of an EM wave interacting with a Hydrogen atom.

When ( )

If

[Check this be evaluating ( ) (

) ]

( )

( )

( )

( )

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(4.7)

∫

∫

∫

(4.8)

Where the 3rd

integral is only non-zero when (Euler)

Done using spherical polar coordinates and previously known solution to the Hydrogen atom.

Thus for all EM waves with , only transitions where are allowed.

[Similarly of ( ) transitions are allowed if ]

Energy-Time Uncertainty Principle

[Figure of spectrum (x-axis) with | | (y-axis), spectrum decays and oscillates at lengths

]

From the graph we can see

(end of lecture, next line is obvious in relating to the Energy-Time Uncertainty Principle

12

Lecture 5: Ehrenfest’s Theorem and the Classical Limit

Concerns the t – dependence of the expectation value,

⟨ ⟩

Recall the Hermitian property: for matrices

For operators

⟨ | | ⟩ ⟨ | | ⟩

(5.1)

So

∫ [∫ ]

∫ ( )

(5.2)

Using the above,

⟨ ⟩ ⟨ ( )| | ( )⟩ ∫ ( ) ( )

⟨ ⟩ ∫

∫ (

) ∫

(5.3)

Time dependence of wave functions tends to be left out to improve visibility.

TDSE:

( )

(5.4)

Thus

⟨ ⟩

∫( )

⟨

⟩

∫

(5.5)

First time is equivalent too

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∫[( )

(

)]

{∫ ( )}

Back into (5.5):

⟨ ⟩

∫ ⟨

⟩

∫

⟨ ⟩

∫ [ ] ⟨

⟩

(5.5)

Note that if [ ] and ⟨

⟩ then

⟨ ⟩

Section 2: The Classical Limit

Basic idea is to examine how Newton’s laws emerge from QM in an appropriate limit.

Identify “particle” with the idea of a “wave packet” {picture of a wave packet}

Use Ehrenfest:

⟨ ⟩

⟨[ ]⟩

(5.6)

Where is momentum

Consider 1 particle 1D,

( )

[ ] [

] [ ( )]

(5.7)

The 1st time is equal to zero, the 2

nd requires some thought:

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Start by considering

[ ( )] ( )

( ) ( ) ( )( )

( )

[ ( )] ( ) ( )

( )

( )( )

( )

(5.8)

The 2nd

and 3rd

terms cancel so

[ ( )] ( ) ( )

[ ( )]

Returning to (5.6)

⟨ ⟩

⟨[ ]⟩

⟨ ⟩ ⟨

( )

⟩

(5.9)

This is the “force” operator.

If the probability distribution functions for and ( )

are sharply peaked on the scale of

experiment, then replace ⟨ ⟩ with “the momentum” and ⟨ ( )

⟩ with “the force”

{Figure, graph showing Gaussian distribution function with on x-axis and the FWHM being }

Condition holds if

We then have Newton’s 2nd

law,

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Lecture 6: Classical Relations

Continuation, recall

(

)(

)

(6.1)

Both brackets can be much smaller than 1, i.e. obeys a classical regime only if

I.e. large or scales

⟨ ⟩

⟨[ ]⟩

(6.2)

[ ( )] [ ] [ ]

[ ]

[ ] ( )

[ ] ( )

[ ]

[ ]

(6.3)

So

⟨ ⟩

⟨

⟩ ⟨

⟩

(6.4)

In the classical limit:

(6.5)

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Wave Packet Evolution

Suppose at { }

Take

( )

( √ )

(

)

(6.6)

{Small illustrations of wave graph diagrams combining into a resultant graph}

Consider free particle ( ( ) ),

Eigen functions are:

( )

Eigen values are:

We wish to express ( ) in terms of these. For completeness, write

( ) ∫

( )

(6.7)

Solution by using inverse Fourier Transform:

( )

∫ ( )

( )

∫

( √ )

(

) ( )

(6.8)

Use standard Fourier Transform of a Gaussian:

17

∫

√

(

)

(6.9)

Where in (6.9)

( )

Therefore

( )

√ ( √ )

(

( )

)

√ ( √ )

(6.10)

So the probability distribution of p is {Figure of [ ( )] }

And taking the theory from lecture 1 (I believe this refers to linear combinations of wave

functions or their solutions):

( ) ∫ ( ( )

) ( )

(6.11)

So

( ) ∫ ( ( )

) ( ) (

)

( ) ∫ ( ( )

) ( ) (

)

( ) ∫ ( ( )

) ( ) (

)

Solved by completing the square in the exponent and using contour arguments, yielding

( ) (

)

( ( )

(

)

)

Lecture 7: Wave broadening and Many Particle Systems

18

Continuation, recall

| ( )|

√

(

(

)

(

))

√

(

(

)

(

))

√

√

(6.1)

[Figure showing this wave and smaller waves that make it up]

Momentum associated with us expect to move a distance

Broadness with time? In terms of wave theory,

(6.2)

is not independent of k broadening i.e. dispersion [Figure showing this on graph]

√

(6.3)

19

Also see hand out on waves, some numerical examples are given highlighting the difference

between classical and quantum systems.

Section 3: Many Particle Systems

E.g. for 2 particles, classically

| |

| |

( )

Replace (

)

( )

( ) ( )

Non-interacting particles

In this case,

( ) ∑ ( )

Then

( )

So can be solved using the method of separation of variables,

( ) ( ) ( ) ( )

Caution; at this stage this is only mathematically correct (See later).

We find the solution is provided,

Then

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∑

I.e. we have N single particle SE’s

Lecture 8: Identical Particles

For identical particles are all the same function i.e. are solutions of the

same 1 particle SE. Mathematical solutions with particles interchanged are degenerate.

E.g. for 2 particles

( ) ( )

( ) ( )

( ) ( )

Separate physical law for identical particles (whether interacting or not)

Generalised Pauli Principle:

must be symmetric w.r.t. exchange of coordinates of any two bosons ( )

must be anti-symmetric w.r.t. exchange of coordinates of any two fermions

( )

E.g.

( ) ( )

( ) ( )

To satisfy this, we must take a particular linear combination of degenerate product wave

functions that have the correct exchange symmetric.

√ ( ( ) ( ) ( ) ( ))

√ ( ( ) ( ) ( ) ( ))

In general,

For Bosons: Symmetrise by adding all permutations of ’s, re-normalize.

For Fermions: Anti-symmetrise by adding all permutations of ’s, with a minus sign for each

exchange, re-normalize.

21

Example, terms being left out for shorthand but are still there

√ [ ]

This anti-symmetrisation (for non inter fermions) can be handled elegantly using a “Slater

Determinant”

√ |

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )|

Recall that any determinate changes sign when rows interchanged.

Note that for non-interacting fermions, if we try to put two particles for the same ( counts

both spatial and spin part of the 1-particle wave function) the value of the anti-symmetrised

is zero (see from S.D. example) – cannot be normalised, so impossible.

So, can’t point fermions in the same 1-particle state (counting spatial + spin state) –

Elementary Pauli Principle

Exchange energy for interacting fermions:

Atom, [Figure showing n energy levels and occupied states by spin]

In absence of interaction,

Lecture 9: Identical Particles Continued

(

)

(

)

| |

| |

Treat pertubatively. In absence of u, where and are eigenvalues of

[Figures of two energy level diagrams showing ground and excited states]

Ground state is

22

( ) ( )

√ ( )

Excited State

√ ( ( ) ( ) ( ) ( ))

{

√ ( )

}

Or*

√ ( ( ) ( ) ( ) ( )

√ ( ))

[Various energy level diagrams, depending on value of ]

Now consider u:

⟨ | | ⟩ ( )

∫| |

(By definition | | and )

is larger for than spatial wave function

So the large strength of (

| |) is felt less strongly

Note [ ] [ ] , so remain good quantum numbers even in the presence of

interaction.

This is known as the “exchange effect” – high spin has lower energy than lower spin [as if

of the same had an attractive interaction].

Review of the Variational Principle:

If

| ⟩ ∑

| ⟩

Where are the exact Eigen functions of

Consider

23

⟨ | | ⟩ ∑∑ ⟨ | | ⟩

| | ⟩ ⟨ | | ⟩

⟨ | | ⟩ ∑| | ( )

∑| |

Note ∑ | |

and | | ( )

⟨ | | ⟩

Error is 2nd

order in ( )

Lecture 10: Hartree Theory

Final points on variational principle;

The ⟨ | | ⟩ is a minimum when , exact ground state wave function

If , error ⟨ | | ⟩ | |

If (say, 7th

Eigen state of ), then ⟨ | | ⟩ is stationary w.r.t. variations in

Approaches to the interacting-fermion problem:

Approach Accuracy Cost

1. Hartree ⟐ £

2. Hartree-Fock ⟐⟐⟐ ££

3. Configuration Interaction ⟐⟐⟐⟐⟐ £££££

4. Density Functional Theory ⟐⟐⟐⟐ ££

Approach Accuracy Cost

Hartree

Hartree-Fock Configuration

Interaction

Density Functional Theory

⟐

⟐⟐⟐

⟐⟐⟐⟐⟐

⟐⟐⟐⟐

£

££

£££££

££

Approaches 1-3 use the variational principle for , the 4th

uses V.P. in terms of ( ).

24

Hartree Theory

Strategy: V.P. for the exact , and product of 1-particle wave functions ( ) ( )

Note, this does not satisfy the generalised Pauli Principle but we gesture towards the P.P. by

insisting that be different eigen functions of some (to be determined) 1 particle

Hamiltonian.

For the case

( )

( )

| |

( ) ( )

( ) ( ) ( )

Calculate

⟨ | | ⟩ ∫ ( )

( ) ( ) ( )

∑

The ( ) term in :

∫

∫ ( ) ( ) ( )

Likewise ( ) yields

∫ ( ) ( ) ( )

Integrated term is

∫∫| ( )| | ( )|

(| |)

Takes reasonable form: average value of u, taking into account electron density | | of

electrons 1 and 2

Suppose that are 2 eigen states satisfying (

)

Now,

25

⟨ | | ⟩ ⟨ | | ⟩ ⟨ | | ⟩ ∫ ( )∫| ( )|

(| |) ( )

The term looks like a potential of

From V.P. this is stationary w.r.t. variations in if satisfies

( )

Similarly for

Hartree potential for electron 1 is:

( ) ∫ | ( )| (| |)

( ) ∫ | ( )|

| |

( )

Compromise (from P.P.) is to choose the

( ) ∫∑| ( )|

| |

I.e. the electrostatic potential at due to charge density of all electrons

So, solve

(

( ) ( )) ( ) ( )

Potential depends on so must solve iteratively (repeat until convergence):

Lecture 11: Hartree-Fock Theory

This combines Hartree theory + exchange energy

The V.P. + trial wave function = anti-symmetrised product function = “Slater Determinate”

E.g.

26

√ ( ( ) ( ) ( ) ( )) ( ) ( )

Construction of ⟨ | | ⟩ follows much as in Hartree theory, except that new terms like

∫ ( ) ( ) (| |) ( ) ( )

Also enter the equation

Once again show that ⟨ | | ⟩ is a minimum provided satisfy a 1 electron S.E.

( ) (

( ) ( )) ( )

( )

∑∫ (

) ( )

(| |)

The 4th

term can be thought of as a non-local operator, exchange operator!

Operating on :

∫∑( ) ( )

Note:

Now exchange operator ( ) is –ve, and arises only from electrons with the same span as .

The unphysical self-interaction energy is, although present in , exactly removed by the

term in the exchange operator.

Once again S.E. can be solved self consistently in a few iterations.

Pretty good for molecules, atoms, etc. with a substantial energy gap between occupied and

unoccupied states (see later on a well-defined Brillion zone).

Hartree-Fock theory omits correlation, i.e. motion of one electron affected by the proximity

of another: can be shown to be stronger when the energy gap is small or zero.

3. Configuration Interaction

In principle exactness relies on completeness of the Slater Determinates made from some

underlying complete set of 1 electron wave functions

Idea:

∑

Optimise Coefficients; to minimise the ⟨ | | ⟩

27

Huge number of configurations makes this feasible up to electrons.

4. Density Functional Theory (DFT)

Based on electron density

( ) ∫| ( )|

Lecture 12: Density Functional Theory

Hohenberg-Kohn Theorem:

If N interacting fermions (usually electrons) move in an external potential ( ) then there

exists a universal function, [ ] such that the functional

( ) [ ] ∫ ( ) ( )

Is minimised when the function, ( ) , the ground state electron density, and [ ]

, the ground state energy of the interacting system. [Figure of [ ] minimising]

Define

∑

∑ (| |)

∑ ( )

Now define

[ ] ⟨ | | ⟩

All N electron ( ( )) ’s are exchange and anti-symmetric and yield the density

function ( ) as defined above. Then

[ ] [ ] ∫ ( ) ( ) [ ] ⟨ |∑ ( )

| ⟩

For each n, let be the that minimises then

[ ] ⟨ | | ⟩ ⟨ |∑ | ⟩

28

[ ] ⟨ | | ⟩

Let be the actual ground state wave function with the density given by

Then from V.P.

⟨ | | ⟩

But also,

⟨ | |

⟩ ⟨ | | ⟩

⟨ | |

⟩ ⟨ | | ⟩

⟨ | |

⟩

⟨ | | ⟩

Kohn-Shan Theory:

We had ( ) [ ] ∫ ( ) ( ) , now write

( ) [ ]

∫∫ ( ) ( ) ( ) [ ] ∫ ( ) ( )

Now [ ] [ ] but without the electron-electron interaction, i.e. the kinetic energy of

non-interacting electron density n

∫∫ ( ) ( ) ( )

[ ]

[ ] [ ] ∫ ( )

( ) ( ) ∫ ( ) ( ) [ ]

( )

( ) ( ) [ ]

( )

So looks like non interacting electrons moving in the potential ( ), so solve

[

]

Then ( ) ∑ | ( )|

, solve self consistently a la Hartree

29

Term needs to be approximated; however it is a fairly small part of the total energy.

Lecture 13: Density Functional Theory Continued

Kohn-Shan:

(

( )) ( ) ( )

( ) [ ]

( )

Usual approximation for :

Local Density Approximation (LDA)

[ ] ∫ ( ( )) ( )

( ) [ ]

( )

Exact if system is HEG, otherwise not too bad

[Figure showing [ ] originating from origin and following ]

DFT can be generalised to the time dependent case to which electrons get excited - TDDFT

4. Second Quantisation

Notation for many particle states

| ⟩

Means particles in etc. where are some convenient complete set of

single-particle wave functions. Implies

30

{

} {

E.g. for fermions,

| ⟩

√ ( ( ) ( ) ( ) ( ))

| ⟩

√ | ( ) ( )

( ) ( )|

Where the matrix is a Slater Determinate. For Bosons,

| ⟩ ( ) ( )

Creation and annihilation operators (note dagger note plus)

| ⟩ | ( ) ⟩

| ⟩ | ( ) ⟩

The sign change is the important difference.

Specific proportionality:

Bosons: √ √ respectively

Fermions: respectively (see later for sign)

Term adds a column, , into the LHS of the Slater Determinate. This fixes the sign

√ |

| |

|

√( )

31

Lecture 14: Annihilation and Commutator Relations

Continuing,

|

|} | ⟩ |

|} | ⟩

Where of swaps (of columns) to bring k to its numerical order, similarly removes

from the LHS of the “Slater Determinate”.

√ |

|

( )

√ |

|

| ⟩ ( )

√( ) |

| | ⟩

This gets quite hard to follow, supplement: Wiki

(Anti) Commutator Relations, Bosons (Commutation) – let

| ⟩ √ √ | ( ) ( ) ⟩

Where as

| ⟩ √

√ | ( ) ( ) ⟩

[

]

Also clearly true if , similarly

[ ]

If

| ⟩

| ⟩

However if ,

| ⟩ √

| ( ) ⟩

Where as

32

| ⟩ √ | ( ) ⟩

√ √ | ⟩ ( )| ⟩

[ ]

Thus in general

[ ]

Fermions (anti-commutation):

√ |

|

√( ) |

| ( )

√( ) |

|

Where as

√ |

|

√( ) |

|

( )

So

{

}

{ }

When ( )

√ |

|

√( ) | |

√ |

|

Where as

√ |

|

√( ) |

|

( )

√ |

|

If

33

√ |

|

√( ) | |

√ |

|

Where as

√ |

|

No column is initial state, we get 0, starting state

Thus always

{ }

Lecture 15: Field Operators

In summary

Bosons:

[ ]

[ ]

[ ]

Fermions:

{

}

{ }

{ }

| ⟩ | ⟩

Field Operators:

Created/annihilation operators that create/destroy particle “ ” (as opposed to

“ ”). Note,

34

∑

( ) ( ) ( )

Then,

( ) ∑ ( )

Creates a particle of with spin . Term is not a function.

Similarly,

( ) ∑ ( )

Destroys a particle of with spin .

(Anti) commutation relations become:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

Define ( ) ( ) as the density operator, number of particles per unit volume.

The in terms of or ( ) ( )

It can be shown (weekly problem week 8/9) that

∑

∑

Where ( )

∫ ( ) ( )

( )

( )

And (check all these subscripts)

∫ ( )

( ) ( ) ( ) ( )

This is equivalent to element in the sense that its matrix element with respect to any p of

many particle states is the same for elementary .

35

For the field operators,

∫ ( ) ( ) ( )

∑∫ ∫ ( ) ( )

( ) ( )

Possibly one more term on the end (check), and highlighting of fermion boson difference

Lecture 16: Heisenberg Picture

(Missed start? Most probably)

Usual Schrödinger picture; represents on observable, no time-dependence when/if the

observable is time-independent, is time-dependent

Heisenberg picture: time-independent, it’s time dependence has been moved to operators

like

TDSE:

( )

( ) ( )

Time dependence of ( ) is only that of ( )

Write ( ) ( ) ( ) where ( ) is the time-evolution operator

( ( )

) ( ) ( ) ( ) ( )

( )

( ) ( )

e.g. if independent of time:

( )

( )

( )

[

]

Physically relevant quantities in QM are the matrix elements of an observable operator, like

⟨ ( )| | ( )⟩ ∫ ( )

36

Where

( ) ( ) ( )

( ) ( ( ) ( ))

( ) ( ) ( )

Where is the complex conjugate, (

)

( ) ( ) ( )

⟨ ( )| | ( )⟩ ∫ ( ) ( ) ( ) ( )

[If , we get ⟨ ( )| ( )⟩ ⟨ ( )| ( ) ( )| ( )⟩ where ( ) ( ) and ]

Thus ( ) ( ) , simarly ( ) ( )

If now identify ( ) ( ) as the wave functions in the Heisenberg picture and ( ) ( )

as the operator in the Heisenberg picture, we have preserved all physical information, i.e.

( ) ( )

(

) (

)

We had

, thus also

( )

(

)

[ ] (

)

[ ]

(

)

[ ] (

)

This is the Equation of Motion of a Heisenberg operator

In the case where , we get

(

)

( ) ( )

Next us equation of motion to get that of

37

Lecture 17: Time-dependence of and operators inducing

field operators

Continuing (or missing some…)

Then,

[ ] (

)

Time-dependence of and operators inducing field operators:

All operators have subscript H and skipping hats for ease of notation.

∑

∑

We have

[ ]

Using (anti) commutation relations of (week problem), it can be shown

[ ]

[

]

So then,

[ ] ∑

∑

{

}

∑

∑

∑

But by symmetry of interaction, , also can swap “dummy” indices. Giving

[ ] ∑

∑

Thus,

38

{∑

∑

}

In the case of field operators ( ) and ( ) we get simplification because,

( )

So we get,

( )

{ ( ) ∑ ( )

( ) ( ) ( )

}

In the absence of interaction, we get

( ) ( )

So the field operators and behave mathematically rather like wave functions in a

single-particle system.

Many body perturbation theory:

One-particle Green’s function (which acts as a propagator),

( ) ⟨ | ( ( ) ( ))| ⟩

( )

Lecture 18: Many body perturbation theory

Continuing (I think)

∫ ( ) ( ) ( ) ( )

( ) ⟨ | [ ( ) ( )]| ⟩

[Figure similar to heat bath systems in TD and Stat Mechanics]

From equations of motion of and can show

39

(

( )) ( ) ∫ ( ) ⟨ | [ ]| ⟩ ( ) ( )

If we define the “self-energy operator” ∑( ) by equating the * term above to “ ∑ ”

i.e.

∫∑( ) ( )

Then we have,

(

( ∑) ( ) ( )

This is also the equation of motion of of a non-interacting “auxiliary” system of electrons,

moving in an effective potential.

( ) ∑( )

Where the ∑ term is a non-local, energy (i.e. time) independent potential

Using this definition of ∑, one can deduce a closed set of coupled equations relating G and ∑

Known as “Hedin’s Equations” (http://arxiv.org/pdf/1109.3972v1.pdf)

One can solve these equations iteratively obtaining ∑ to desired order in (e.g.)

(E.g. The first order term in for ∑ is )

Quantisation of fields (QED):

Define

( )

( )

}

( )

Similarly ( ) ( )

The field operators, linear combinations of or ( ) create (or destroy) photon at with

specific polarisation.

( )

40

( )

And

( ) ( )

Is the field operator (obeys the same wave equations as classical ) which in the classical

limit reduces to ( ), the vector potential.

Can combine with to yield a quantum theory of fields and matter – QED

Fin