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 1
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 2
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. The solution is: (1.5) (1.6) So for each Eigen function that contributes to the wave function ( ( own rate. )) oscillates at its Taking the example of the harmonic oscillator: { } (1.8) Examining the { } part of eqn (1.8) implies 3
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 2 nd term is assumed to be small Write (2.2) Where can be replaced with And 4
In the expectation that for weak, the functions will be slowly varying (c.f. case (b)) TDSE Using TDSE, { } { } Where two terms clearly cancel (2.3) 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 5
(2.7) (2.8) Write And so (2.9) This (2.9) is 1 st order time-dependent perturbation theory; valid provided too much from zero, i.e. has not declined too much from 1. has not increased 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) 6
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, 1 st 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 7
(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, () {Check why the time term leaves the sin argument looks like mistake in (3.9) or (3.10)} (3.10*) ( ) 8
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 9
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 ] 10
(4.7) (4.8) Where the 3 rd 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 11
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 12
[( ) ( )] { ( )} 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 1 st time is equal to zero, the 2 nd requires some thought: 13
Start by considering [ ] [ ] (5.8) The 2 nd and 3 rd 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 2 nd law, 14
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) 15
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: 16
(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): ( ) So (6.11) ( ) ( ) ( ) Solved by completing the square in the exponent and using contour arguments, yielding ( ) Lecture 7: Wave broadening and Many Particle Systems 17
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) 18
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 19
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: E.g. 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 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. 20
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 21
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 22
Note and Error is 2 nd 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, 7 th 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 4 th uses V.P. in terms of. 23
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 electrons 1 and 2 of Suppose that are 2 eigen states satisfying Now, 24
() 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. 25
( ) 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 4 th 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 26
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 exists a universal function, [ ] such that the functional then there [ ] 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 [ ] 27
[ ] 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 28
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 29
{ } { 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 30
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 31
[ ] Thus in general [ ] Fermions (anti-commutation): Where as So { } { } When Where as If 32
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 ). Note, (as opposed to 33
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. 34
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: like time-independent, it s time dependence has been moved to operators 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 35
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 36
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, 37
{ } 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 single-particle system. in a 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 38
( ) [ ] 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 moving in an effective potential. of a non-interacting auxiliary system of electrons, 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. 39
And Is the field operator (obeys the same wave equations as classical limit reduces to, the vector potential. ) which in the classical Can combine with to yield a quantum theory of fields and matter QED Fin 40