Advanced Quantum Theory

Problems

Sebastian Muller (University of Bristol)

2020/21, teaching block 2

The homework for Advanced Quantum Theory will be set from this handout, from Wednesdays toWednesday in the following week at 3 pm.

Set homework will be announced in the lecture as well as on the course webpage. The set questionfor the first homework, due on 10 February, is 2.1. The solutions will be added to the solutionfile on Blackboard after the corresponding homework deadlines.

This handout also contains past exam questions for the course, and a few selected solutions to questionsthat I am not planning to set as homework (e.g. for students who want to see additional solved problemsbefore starting their homework).

All course material is provided for educational purposes at the University of Bristol and is to be down-loaded or copied for your private study only.

1 Basics

1.1 Stationarity of the action in Hamiltonian mechanics

The analogue of the action in Hamiltonian mechanics is

S[q,p] =

∫ t2

t1

(p(t′) · q(t′)−H(q(t′),p(t′))dt′

where H(q,p) is the Hamiltonian of the system. Using the Euler-Lagrange equations from Mechanics2/23 determine the conditions under which S[q,p] is stationary w.r.t. variations of the functions q(t)and p(t) that preserve the boundary conditions at t1 and t2.

1.2 A representation of the delta function

Show that1

2π~

∫ ∞−∞

eipx/~dp = δ(x).

Possible ways to solve this problem are

(a) Consider1

2π~lima→0

∫ ∞−∞

eipx/~−ap2dp

and use that

limε→0

1√πεe−y

2/ε = δ(y).

1

(b) Use that subsequent application of the Fourier transform and the inverse Fourier transform to afunction returns the function we started from.

2 Feynman path integral

2.1 Path integral in phase space

Show that the propagator of a quantum system can be written as

K(rf , r0, t) =

∫D[r]D[p] exp

(i

~

∫ t

0(p(t′) · r(t′)−H(r(t′),p(t′))dt′

)Here the integral is taken over all functions r(t′) with r(0) = r0 and r(t) = rf , and over all functionsp(t′) regardless of boundary conditions.To solve this problem use the result

K(rf , r0, t) =

∫dnr1 . . .

∫dnrN−1

N−1∏j=0

〈rj+1|e−i~ Hτ |rj〉

from the lecture (with rN := rf ), as well as an expression for 〈rj |e−i~ Hτ |rj−1〉 where the integral

over the momentum has not been evaluated, such that momenta for the different time steps remain asintegration variables in the final result.

Note that the integration measure will be different from the D[r] appearing in the position-space pathintegral. Your calculation should give the appropriate meaning of

∫D[r]D[p] . . . .

2.2 Propagator for the harmonic oscillator

In the lecture we determined the propagator for the harmonic oscillator. Check explicitly that theobtained result satisfies the Schrodinger equation(

− ~2

2m

∂2

∂x2f+

1

2mω2x2f

)K(xf , x0, t) = i~

∂

∂tK(xf , x0, t)

and that it satisfies K(xf , x0, 0) = 〈xf |x0〉 = δ(xf − x0). It may be helpful to use

limε→0

1√iπε

eiy2/ε = δ(y).

2.3 Propagator for a free particle

Evaluate the path integral for the propagator of a particle moving freely in one dimension withoutpotential, i.e., a particle with the Lagrangian L = 1

2mx2.

Hints: As for the harmonic oscillator, split x(t′) into the classical solution xcl(t′) and the deviation

δx(t′) from this solution. Then work with the discretised version of the position-space path integral,and show that the action S[δx] can be written as

S[δx] = δx ·Aδx

where δx is a vector whose components are the values of δx at the time steps and A is a real symmetricmatrix. Moreover you can use that for such a matrix we have∫ ∞

−∞

∫ ∞−∞

. . .

∫ ∞−∞

exp (iδx ·Aδx) dδx1dδx2 . . . dδxν =

((iπ)ν

detA

)1/2

.

2

(how do we have to choose ν for our problem?) and that the ν × ν matrix of the form

Bν =

2 −1 0 · · · 0−1 2 −1 · · · 00 −1 2 · · · 0...

......

. . . −10 0 0 −1 2

has the determinant detBν = ν + 1.

2.4 Free particle in Hamiltonian mechanics

Evaluate the Hamiltonian version of the path integral for the propagator of a particle moving freely in

one dimension without potential, i.e., a particle with the Hamiltonian H = p2

2m .

2.5 Short-time propagator [2019 exam]

We consider a one dimensional system whose potential U is an arbitrary function of the position x andwhose kinetic energy T is an arbitrary function of the momentum p. Show that for short times t thecorresponding propagator can be approximated by

K(xf , x0, t) ≈1

2π~

∫ ∞−∞

dp exp

(i

~

[pxf − x0

t− T (p)− U(x0)

]t

)where the two sides agree neglecting corrections of order t2 and higher.

2.6 Path integral with products of position and momentum [2017 exam]

We consider a quantum mechanical system in one dimension with the Hamiltonian

H =1

2p2 +

1

2(px+ xp)

where x is the position operator and p is the momentum operator.

(a) Show that the propagator of the system can be written as a path integral

K(xf , x0, t) =

∫D[x]D[p]e

i~S[x,p]−

t2

where the integral runs over phase space trajectories x(t′), p(t′) with x(0) = x0 and x(t) = xf .S[x, p] is the classical action associated to the phase space trajectory, and t is an arbitrary positivetime. Your answer should give a definition of the the integration measure

∫D[x]D[p] . . . used in

the formula. You can use without proof the following result for small times

K(xf , x0, t) =1

2π~

∫ ∞−∞

dp exp

(− i~

(1

2p2 + xfp

)t− t

2+i

~p(xf − x0)

)+O(t2). (1)

(b) Using your answer to (b)i show that for the system in question we have

K(xf , x0, t) = A(t) exp

(i

~

∫ t

0

1

2p(t′)2dt′

)where p(t′) is the momentum evaluated on a classical trajectory with initial and final positionsx(0) = x0 and x(t) = xf and A(t) is independent of x0 and xf . You are not asked to determinethis trajectory or evaluate the integral.

(c) Now give a proof of the result in Eq. (1).

3

2.7 Derivation of the Statistical Mechanics path integral [2018 exam]

(a) Show that for a one-dimensional quantum system with Hamiltonian operator H the matrix ele-

ments of e−βH can be written as the path integral

〈xf |e−βH |x0〉 =

∫D[x]D[p] exp

(i

∫ β

0p(β′)x(β′)dβ′ −

∫ β

0H(x(β′), p(β′))dβ′

).

Here ~ and the mass are set to 1, β is a positive real number, the dot indicates a derivativew.r.t. β′, and H is the classical Hamiltonian. The integral runs over all functions x(β′) (where0 ≤ β′ ≤ β) with x(0) = x0 and x(β) = xf , and over all functions p(t′). The form of D[x]D[p]should follow from your derivation.

In your derivation you may restrict yourself to systems with a quantum Hamiltonian H = T + Uwhere T = 1

2 p2 and U is the potential depending only on x. You are not permitted to use the

path integral for the propagator derived in the lecture.

(Basically, this question asks to derive the phase-space Statistical Mechanics path integral in away similar to question 2.1 (a).)

(b) Using the intermediate results of (a) or otherwise, also derive the path integral

〈xf |e−βH |x0〉 =

∫D[x] exp (−SE [x])

where the Euclidian action is given by

SE [x] =

∫ β

0dβ′(

1

2x(β′)2 + U(x(β′))

)and the form of D[x] should follow from your derivation. You may use without proof the followingformula (which holds for a > 0) ∫ ∞

−∞e−ay

2+icydy =

√π

ae−

c2

4a .

2.8 Elastic chain [not examinable]

We consider a chain of N+1 labelled by indices i = 0, . . . , N . As in the lecture masses with neighbouringindices are connected by a spring with spring constant k and natural length zero. In contrast to thelecture the positions of the masses are taken as two-dimensional. The position of the i-th mass isdenoted by Φi = (Φi1,Φi2). The mass i = 0 is fixed at Φ01 = Φ02 = 0, the final mass with index i = Nis fixed at ΦN1 = C,ΦN2 = 0. The masses have a kinetic energy, a potential energy due to the tensionof the springs, and also a gravitational energy mgΦi2 for each mass.

(a) Write the propagator of the system as a path integral. (You don’t need to evaluate the pathintegrals in this question.)

(b) Then take the limit N → ∞ of this path integral, choosing the positions Φi1 for the case ofequidistant masses as a continuous parameter replacing the index i.

(c) Write down a path integral analogous to (b) for the matrix elements of e−βH .

4

2.9 Euler-Lagrange equations for the continuous case [not examinable]

(a) Consider a functional acting on functions r(x, t′) where x and t′ are real numbers and r is in Rn.The functional is defined through the integral

S[r] =

∫ t

0dt′∫ C

0dx L(r(x, t′), r′(x, t′), r(x, t′))

where r′ = ∂r∂x , r = ∂r

∂t′ . Show that S[r] is stationary w.r.t. variations of r(x, t′) (that preservethe boundary conditions at x = 0, C and t′ = 0, t if

∂L∂r

=∂

∂x

∂L∂r′

+∂

∂t′∂L∂r

.

For the proof you can proceed similarly to the proof of the Euler-Lagrange equations in Mechanics2/23.

(b) Using (a) give an example for a function Φ(x, t′) for which the action of the elastic chain in chapter2.4 is stationary.

(c) Determine the Euclidian action for the elastic chain, and give an example for a function Ψ(x, t′)for which this Euclidian action is stationary.

2.10 A two dimensional grid with springs [modified from 2016 exam but not ex-aminable this year]

Consider a two-dimensional grid of (N + 1)2 masses m connected by springs (as sketched in the figurefor the case N = 3). The masses are numbered by integers i = 0 . . . N and j = 0 . . . N and their

positions are denoted by Rij =

(Xij

Yij

). The masses whose indices i differ by 1 are connected by a

spring, and the same holds for masses whose indices j differ by 1. The springs have the spring constantk and the natural length 0. The positions of the masses with i = 0 and i = N are fixed according to

R0j =

(0jN

), RLj =

(1jN

), and the remaining masses are allowed to move freely.

(a) Write down the Lagrangian and the action of the system. You are not asked to take into accountgravity.

(b) Show that in the continuum limit L → ∞, Rij(t′) can be replaced by a field R(x, y, t′) and the

Lagrangian from (b)i can be written as a multiple integral over a Lagrangian density that dependson R(x, y, t′) and/or its first derivatives w.r.t. x, y, and t′.

3 Perturbation theory

3.1 Feynman’s trick

(a) Express the integrals

I2n =

∫ ∞−∞

x2ne−ax2/2dx

5

(where n is a natural number) through derivatives of∫ ∞−∞

e−ax2/2dx (2)

w.r.t. a.

(b) Use the expression from (a) as well as the result for (2) to determine I2n. This idea is similar to(but slightly less powerful than) the method we will use to evaluate these integrals in the lecture.

3.2 Complex Gaussian integral

(a) Evaluate the integral ∫ ∞−∞

∫ ∞−∞

e−a|z|2dRe z d Im z

(for a > 0) and, using this result, determine

c ≡∫Cn

e−z†Az

n∏k=1

dRe zk d Im zk

where z ∈ Cn and A is a hermitian (self-adjoint) positive definite n × n matrix. To answer thisquestion it is helpful to express A in terms of a diagonal matrix, a unitary matrix and its adjoint.

(b) We now consider the average

〈. . .〉 ≡ 1

c

∫Cn

e−z†Az . . .

n∏k=1

dRe zk d Im zk.

Show that 〈zkzk′〉 = 〈z∗kz∗k′〉 = 0 and evaluate zkz∗k′ . This can be done by considering a generating

function 〈ej†z+z†j〉 and taking derivatives w.r.t. this generating function treating jk and j∗k asindependent variables.

3.3 Wick’s theorem

Use Wick’s theorem to evaluate the following integrals:

(a)∫

(x+ y + z)2e−10x2−y2−6xy−2z2dxdydz

(b)∫xixje

− 12xTAx−ε

∑k x

4kdnx neglecting terms of order ε2 and higher, for real symmetric positive

definite n× n matrices A and i, j being integers between 1 and n

(c)∫e−

12xTAx−ε(x41+x42)d2x neglecting terms of order ε3 and higher, for x ∈ R2 and real symmetric

positive definite 2× 2 matrices A

3.4 Wick’s theorem continued [solved in end of booklet]

Use Wick’s theorem to evaluate the integral∫e

i2xTAx+iε

∑k x

3kdnx neglecting terms of order ε3 and

higher, for real symmetric invertible n× n matrices A.

6

3.5 Matrix integrals

We are considering real symmetric 2× 2 matrices H and we define

〈. . .〉 ≡ c−1∫e−trH

2. . . dH11 dH12 dH22

where

c ≡∫e−trH

2dH11 dH12 dH22 .

Use Wick’s theorem to evaluate

(a) 〈det(H − E)〉 where E is a real number

(b) 〈HαβHγδ〉

(c) [Not examinable] Write down the possible contractions for

〈trH4〉 =∑αβγδ

〈HαβHβγHγδHδα〉.

For each way of drawing contraction lines, which conditions do the summation variables α, β, γ, δhave to satisfy to give nonvanishing contributions to the average?

3.6 Integral kernel

In the lecture we looked for the kernel G(t′, t′′) subject to the conditions(−m

~

)( ∂2

∂t′2+ ω2

)G(t′, t′′) = δ(t′ − t′′)

andG(0, t′′) = G(t, t′′) = 0.

We gave a formula for this kernel and proved it. Now imagine that you don’t know this result anddetermine G(t′, t′′) from the above formulas by calculation. Consider

(a) the case ω = 0 corresponding to a free particle, and

(b) the case of arbitrary ω 6= 0.

3.7 Propagator for the anharmonic oscillator

Evaluate the integral∫ t0 G(t′, t′)2dt′ for the anharmonic oscillator omitted in the lecture. Use this result

to determine the propagator K(0, 0, t) ignoring terms of order ε2 and beyond.

3.8 Ground state of the anharmonic oscillator

We want to determine the ε2 correction to the ground state energy of the anharmonic oscillator.

(a) First evaluate iG(~i β′, ~i β

′′) for general β′, β′′ between 0 and β, assuming that β is large. (It is okif your approximation breaks down for the case that one of the variables is very close to 0 or β.)You should obtain

iG

(~iβ′,

~iβ′′)≈ ~

2mωe−~ω|β

′−β′′|.

7

(b) Use the result of (a), as well as the results about the anharmonic oscillator for the lecture, to

determine the contribution to 〈0|e−βHanh |0〉 from the Feynman diagram with multiplicity 24.

(c) Evaluate the remaining diagrams and thus determine the ε2 contribution to the ground stateenergy.

3.9 General initial and final positions

In the lecture we have used perturbation theory to determine the propagator of the anharmonic oscillatorfor xf = x0 = 0. Explain how one would obtain the propagator for general xf and x0. It is sufficient toexplain the method and what changes for general initial and final conditions, you don’t need to carrythrough the calculations.

3.10 Feynman diagrams

Use perturbation theory to evaluate the following expressions, and draw the corresponding Feynmandiagrams. Here the meaning of 〈. . .〉 is the same as in the lecture about the anharmonic oscillator. Yourresult should involve integrals over products of factors iG(t′, t′′), and you don’t need to evaluate theseintegrals explicitly.

(a)⟨

exp(− iε

~∫ t0 dt

′x(t′)3)⟩

neglecting terms of order ε4 and higher

(b)⟨x(t1)x(t2) exp

(− iε

~∫ t0 dt

′x(t′)6)⟩

neglecting terms of order ε2 and higher

(c)⟨x(t1)x(t2) exp

(− iε

~∫ t0 dt

′x(t′)3)⟩

neglecting terms of order ε3 and higher

3.11 Feynman diagrams continued [solved in end of booklet]

Proceed as in the previous question for⟨x(t1)x(t2) exp

(− iε

~∫ t0 dt

′x(t′)4)⟩

neglecting terms of order ε2

and higher.

3.12 Quintic perturbation [2017 exam]

(a) We consider the average defined by

〈. . .〉 =

∫Rn

1

cexp

(i

2xTAx

). . . dnx

where A is a real symmetric matrix whose eigenvalues are different from zero, x is a vector in Rnand the constant c is chosen such that the average is normalised. (You are not asked to determinec.) State how averages 〈xk1xk2xk3 . . .〉 can be evaluated using Wick’s theorem.

(b) The harmonic oscillator with a quintic perturbation has the Lagrangian

L =1

2mx2 − 1

2mω2x2 − εx5

where ε is a real parameter much smaller than 1. Use the continuous version of Wick’s theorem towrite down an approximation for the propagatorK(0, 0, t) of this system (here we set x0 = xf = 0),neglecting all terms of order ε3 and higher. Your result should be written in terms of the propagatorfor the unperturbed harmonic oscillator Kharm(0, 0, t) and integrals involving the function G(t′, t′′)

8

introduced in the course as the integral kernel of the continuous analogue of A−1. You do nothave to give the explicit formulas for Kharm(0, 0, t) or G(t′, t′′), and you do not have to evaluatethe integrals. Also draw the Feynman diagrams associated to your result.

(c) Upon replacing i~ t by β the propagator K(0, 0, t) turns into 〈0|e−βH |0〉. You may take as given

that upon a similar replacement iG turns into

iG

(~iβ′,

~iβ′′)≈ ~

2mωe−~ω|β

′−β′′|

where the approximation holds for large 0 < β′, β′′ < β. Use this formula and the result of(b) to determine how much the quintic perturbation would change the ground state energy of theharmonic oscillator if among the Feynman diagrams identified in (b) we consider only the diagramwith the second-largest multiplicity.

(Note: One can show that the harmonic oscillator with a quintic perturbation has additional lowenergy states inaccessible from perturbation theory but you are not asked to consider this issuein your answer.)

(d) We now consider a perturbed harmonic oscillator with the Lagrangian

L =1

2mx2 − 1

2mω2x2 − εxn

where n is an odd number with n ≥ 3. Which are the Feynman diagrams responsible for theleading non-vanishing perturbation to the propagator, and what are their multiplicities? You arenot asked to evaluate any of these Feynman diagrams or write down the underlying integrals.

3.13 Sextic perturbation and quartic prefactor [2018 exam]

(a) We consider the average defined by

〈. . .〉 =

∫Rn

1

cexp

(i

2xTAx

). . . dnx

where A is a real symmetric matrix whose eigenvalues are different from zero, x is a vector in Rnand the constant c is chosen such that the average is normalised. (You are not asked to determinec.) State how averages 〈xk1xk2xk3 . . .〉 can be evaluated using Wick’s theorem.

(b) The harmonic oscillator with a sextic perturbation has the Lagrangian

L =1

2mx2 − 1

2mω2x2 − εx6

where ε is a real parameter much smaller than 1. Use the continuous version of Wick’s theoremto evaluate the path integral

I =

∫D[x]

(∫ t

0dt′′ x(t′′)4

)exp

(i

~

∫ t

0dt′ L(x(t′), x(t′))

), (3)

neglecting all terms of order ε2 and higher. Here∫D[x] . . . indicates a path integral over all

x(t′) such that x(0) = x(t) = 0. Your result should be written in terms of the propagator forthe unperturbed harmonic oscillator Kharm(0, 0, t) for initial and final position 0, and integralsinvolving the function G(t′, t′′) introduced in the course as the integral kernel of the continuousanalogue of A−1. You do not have to give the explicit formulas for Kharm(0, 0, t) or G(t′, t′′), andyou do not have to evaluate the integrals. Also draw the Feynman diagrams associated to yourresult.

9

(c) We now consider a perturbed harmonic oscillator with the Lagrangian

L =1

2mx2 − 1

2mω2x2 − εxn

and the path integral

I =

∫D[x]

(∫ t

0dt′′ x(t′′)p

)exp

(i

~

∫ t

0dt′ L(x(t′), x(t′))

)(4)

where we have n, p ≥ 3 and n and p are either both even or both odd.

(i) Which are the Feynman diagrams responsible for the term proportional to ε in I as definedin (4), and what are their multiplicities? You are not asked to evaluate these Feynmandiagrams.

(ii) Which number do the multiplicities have to sum to? Check explicitly that the multiplicitiesyou obtained sum to this value for the case p = 4 (where n is an arbitrary even number largeror equal to 4).

3.14 Cubic perturbation [2019 exam]

(a) We consider the average of a function f(x) defined by

〈f(x)〉 =

∫Rn

1

cexp

(−1

2xTAx

)f(x)dnx

where A is a positive definite real symmetric matrix, x is a vector in Rn and the constant c ischosen such that the average is normalised. You may take as given that⟨

ejTx⟩

= exp

(1

2jTA−1j

)holds for arbitrary j ∈ Rn. By taking derivatives of this result, evaluate the average 〈xkxk′〉.

(b) We consider a harmonic oscillator with a cubic perturbation

L =1

2mx2 − 1

2mω2x2 − εx3

where ε is a real positive parameter much smaller than 1. Use the continuous version of Wick’stheorem to evaluate the path integral

I =

∫D[x] x(t1)

(∫ t

0dt2 x(t2)

2

)exp

(i

~

∫ t

0dt′ L(x(t′), x(t′))

), (5)

neglecting all terms of order ε2 and higher. Here t1 is a time between 0 and t and is not integratedover.

∫D[x] . . . indicates a path integral over all x(t′) such that x(0) = x(t) = 0. Your result

should be written in terms of the propagator for the unperturbed harmonic oscillator Kharm(0, 0, t)for initial and final position 0, and integrals involving the function G(t′, t′′) introduced in the courseas the integral kernel of the continuous analogue of A−1. You do not have to give the explicitformulas for Kharm(0, 0, t) or G(t′, t′′), and you do not have to evaluate the integrals. Also drawthe Feynman diagrams associated to your result.

(c) We now consider a perturbed harmonic oscillator with the Lagrangian

L =1

2mx2 − 1

2mω2x2 − εx2k+1

where k is an arbitrary natural number larger or equal to 1. For this Lagrangian we are alsointerested in the integral I defined in Eq. (5). Which are the Feynman diagrams responsible forthe term proportional to ε in I as defined in Eq. (5), and what are their multiplicities?Also check explicitly that the multiplicities sum to a double factorial.You are not asked to evaluate the Feynman diagrams.

10

4 Second quantisation

4.1 Three particles in different states

(a) Consider a three-particle system in which we know that the three (different) single-particle states|i〉, |j〉, and |k〉 are occupied by one particle each. Write this state as a linear combination ofsuitable basis states, such as |i〉|j〉|k〉 and |i〉|k〉|j〉. Note that at this stage we do not yet requirethe particles to be indistinguishable.

(b) Then assume that the particles are indeed indistinguishable and bosonic. Which relation doesthis imply for the coefficients in your linear combination? Show that this relation is equivalent tothe result given in the lecture.

(c) Then do the same for fermionic particles.

(d) Your results in (b) and (c) should give you the multi-particle state up to a normalisation factor.Determine this factor.

4.2 Three particles and a double-occupied state

Consider a bosonic system with three particles. Two of these particles are in the state |i〉 and one is inthe state |j〉 with |j〉 6= |i〉.

(a) The normalised three-particle state |i, i, j〉 must be of the form

|i, i, j〉 = α|i〉|i〉|j〉+ β|i〉|j〉|i〉+ γ|j〉|i〉|i〉.

Determine α, β and γ using that the system is bosonic and that |i, i, j〉 is normalised.

(b) Now determine |i, i, j〉 using the general formula for bosonic states from the lecture, and showthat the result is equivalent to (a).

4.3 Wavefunctions and occupation number representation [Exam 2019]

Consider a fermionic (and spinless) system with two single-particle states. Determine the wavefunctionsassociated to the states that in occupation number representation are denoted by |1, 1〉 and by 1√

2(|1, 0〉+

|0, 1〉). In your answer denote by ψi(r) the wavefunction associated to the single-particle state i.

4.4 Relation between ai and a†i

Show that if |Ψ〉 and |Φ〉 are two states with arbitrarily many indistinguishable bosonic particles and

ai, a†i are creation and annihilation operators, we have

〈Φ|ai|Ψ〉 = 〈Ψ|a†i |Φ〉∗

i.e. ai and a†i are adjoint operators. In your proof you can use that the scalar product of states withdifferent particle numbers vanishes by definition.

11

4.5 A state

For a system with indistinguishable bosonic particles, determine the state

∏i

(a†i )mi

√mi!|0〉

in occupation number representation. Here |0〉 is the vacuum, and mi are integers.

4.6 Bose-Hubbard model

We consider a system allowing for an arbitrary number of indistinguishable bosonic particles. Thesystem has two sites, and the operators a†i and ai (i = 1, 2) are the creation and annihilation operatorsfor particles on these sites. The Hamiltonian for the system is

H = −a†1a2 − a†2a1 +

1

2

∑i

ni(ni − 1)

where ni = a†iai. Here the first two terms are related to particles hopping between the two sites, andthe final term represents an interaction potential arising if particles share the same site. This system isa variant of the Bose-Hubbard model which is widely used in condensed matter theory.

(a) Show that applying this Hamiltonian to a state does not change the overall number of particlesinvolved in that state. (To show this you can for example apply the Hamiltonian to a state inoccupation number representation.)

(b) Now we are interested specifically in the case that our system contains two particles. Write downthe basis states of the system with two particles.

(c) Investigate how the Hamiltonian acts on the basis states determined in (b). Use your resultto represent the Hamiltonian (now restricted to the case of two particles) in matrix form anddetermine its eigenvalues.

4.7 Two bosons [2017 exam]

Consider a bosonic (and spinless) quantum system with two single-particle states and the Hamiltonian

H = −2∑j=1

a†j

2∑k=1

ak +2∑j=1

a†j2a2j .

Here aj is the annihilation operator for particles in the state j and a†j is the corresponding creationoperator. In occupation number representation these operators are defined by

a†j | . . . , nj , . . .〉 =√nj + 1| . . . , nj + 1, . . .〉

aj | . . . , nj , . . .〉 =√nj | . . . , nj − 1, . . .〉.

Determine all energy levels for the case that two particles are present in the system.

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4.8 Hubbard model without spin

We consider a system allowing for indistinguishable fermionic particles. The system has three sites, andthe operators a†i and ai (i = 1, 2, 3) are the creation and annihilation operators for particles on thesesites. The Hamiltonian of the system is

H = −a†1a2 − a†2a3 − a

†2a1 − a

†3a2 +

1

2

∑i

ni(ni − 1)

where ni = a†iai. This is the fermionic counterpart of the Bose-Hubbard model studied in question 1but we are interested in the case of three sites.

(a) Explain why in the present case the interaction term 12

∑i ni(ni − 1) does not matter.

(b) We are interested specifically in the case that our system contains two particles. Write down thebasis states of the system with two particles.

(c) Investigate how the Hamiltonian acts on the basis states determined in (a). Use your resultto represent the Hamiltonian (now restricted to the case of two particles) in matrix form anddetermine its eigenvalues.

4.9 Hubbard model with spin

Consider the fermionic system with the Hamiltonian

H = −a†1a2 − a†2a1 − a

†3a4 − a

†4a3 + U(n1n3 + n2n4)

where ni = a†iai. Here the states 1 and 2 are spin-up states at the first and the second site, and thestates 3 and 4 are spin-down states at the first and the second site. The Hamiltonian contains termsrelated to jumps between the two sites, and interaction terms that contribute if both the spin-up stateand the spin-down state at the same site are occupied.

Determine the energy levels of the system for the case that three particles are present.

4.10 Arbitrarily many sites [2016 exam]

(This was following a question dealing with the special case K = 3 so you may want to consider K = 3 toget an idea for a solution.) Consider a fermionic (and spinless) quantum system with K single-particlestates and the Hamiltonian

H = −K−1∑i=1

(a†i+1ai + a†iai+1)−U

2N(N − 1)

where N =∑K

i=1 a†iai and U is a real number. We are interested in the energy levels for the case that

K − 1 particles are present in the system. Determine a matrix H such that all these energy levels areeigenvalues of H. You are not asked to evaluate the energy levels explicitly.

4.11 ai and a†i for fermions

(a) Show that the creation and annihilation operators for fermions ai, a†i satisfy the anticommutation

relations as given in the lecture.

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(b) Show that if |Ψ〉 and |Φ〉 are two states with arbitrarily many indistinguishable fermionic particles

and ai, a†i are creation and annihilation operators, we have

〈Φ|ai|Ψ〉 = 〈Ψ|a†i |Φ〉∗

i.e. ai and a†i are adjoint operators. In your proof you can use that the scalar product of stateswith different particle numbers vanishes by definition.

4.12 Fermions with three single-particle states [2018 exam]

Consider a fermionic (and spinless) quantum system with three single-particle states and the Hamilto-nian

H = −a†1a2 − a†2a1 − a

†3a2 − a

†2a3 +

U

2N(N − 1)

where U is a real number and we have N =∑3

i=1 a†iai. Here ai is the annihilation operator for particles

in the state i and a†i is the corresponding creation operator. In occupation number representation theseoperators are defined by

a†i | . . . ni = 0 . . .〉 = (−1)∑i−1

j=1 nj | . . . ni = 1 . . .〉a†i | . . . ni = 1 . . .〉 = 0

ai| . . . ni = 0 . . .〉 = 0

ai| . . . ni = 1 . . .〉 = (−1)∑i−1

j=1 nj | . . . ni = 0 . . .〉.

How many energy levels does this system have? (For this answer any degenerate levels should be countedseveral times according to their multiplicity.) Determine all these energy levels, except those involvingtwo particles.

4.13 Fermions with two single-particle states [2019 exam]

Consider a fermionic (and spinless) system with two single-particle states and the Hamiltonian

H = 2a1a2 + 2a†2a†1 − a

†1a2 − a

†2a1 .

Here ai is the annihilation operator for particles in the state i and a†i is the corresponding creationoperator. In occupation number representation these operators are defined by

a†i | . . . ni = 0 . . .〉 = (−1)∑i−1

j=1 nj | . . . ni = 1 . . .〉a†i | . . . ni = 1 . . .〉 = 0

ai| . . . ni = 0 . . .〉 = 0

ai| . . . ni = 1 . . .〉 = (−1)∑i−1

j=1 nj | . . . ni = 0 . . .〉.

(a) Show by considering the state |0, 0〉 that application of H to a state may change the number ofparticles in this state.

(b) Determine all energy levels of the system.

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4.14 Fermionic system with arbitrarily many sites [2019 exam]

Consider a fermionic (and spinless) system with L single-particle states and the Hamiltonian

H = −L∑i=1

(a†i+1ai + a†i−1ai)

where ai and a†i are defined as in (b) for i = 1, 2, . . . , L and we formally set a†0 = a†L, a†L+1 = a†1. Deter-mine a matrix such that all energy levels corresponding to states with a single particle are eigenvaluesof this matrix. Determine one of these energy levels as well as the corresponding eigenstate of theHamiltonian.

5 Path integral in second quantisation

5.1 Path integral for systems in second quantisation

In the lecture we considered the following path integral for a system with discrete sites in secondquantisation

tr e−i~ Ht =

∫D[a1, a2, . . .] exp

[ ∫ t

0dt′(−∑j

a∗j (t′)aj(t

′)

− i~H(a1(t

′), a2(t′), . . . , a∗1(t

′), a∗2(t′), . . .)

)].

Here the integral is taken over functions aj(t′) that satisfy the condition aj(0) = aj(t).

(a) [Not examinable] Specify the integration measure∫D[a1, a2, . . .] . . . based on the analogy to

the harmonic oscillator discussed in the lecture.

(b) Write down the path integral for tr e−i~ Ht for the many-particle Hamiltonian H =

∑i,j h(i, j)a†iaj+

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∑i,j Uint(i, j)a

†ia†jajai.

(c) [Not examinable] Assume that the creation and annihilation operators are for states fixed atdiscrete sites with positions xj = j

K . Here the index j runs from 0 to K− 1 and we have periodicboundary conditions meaning that the K-th site at xK = 1 is identified with the 0-th site atx0 = 0. Also assume that the matrix with the elements h(i, j) has the form

h = − ~2

2mK2

−2 1 11 −2 1

1. . .

. . .. . .

. . . 11 1 −2

where all elements apart from those on the diagonal, next to the diagonal, and in the upper rightand lower left corners are zero. This form arises from discretising the kinetic energy in the caseof sites with distances 1

K . Moreover we take Uint(i, j) = 0 i.e. there is no interaction. For thissetting determine the path integral in (b) in the continuum limit K →∞. You should obtain anintegral involving a(x, t′) and a∗(x, t′) that still has a factor K in the exponent.

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5.2 A Grassmann integral [not examinable]

Evaluate the integral∫ 3∏j=1

(dη∗jdηj)(η∗1 + η∗2 + η∗3)(η1 + η2 + η3) exp(2iη∗1η2 − 2iη∗2η1 + η∗3η3)

where ηj , η∗j are Grassmannians.

5.3 Complex numbers and Grassmannians [not examinable]

Derive a formula for tr(λI − A)−1 (where A is a hermitian matrix) involving a complex integral, aGrassmannian integral, and a derivative w.r.t. λ. This problem can be solved by using det = exp tr lnas well as the Gaussian integrals for complex numbers and Grassmannians. It is useful as a startingpoint if one wants to compute averages of such traces.

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3.4 Wick’s theorem continued

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3.11 Feynman diagrams continued

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