preliminaryexercises

Seventh week: relativistic quantum mechanics

Reading assignment for Wednesday April 29th, for class at 11 am:

Interpretation of the Dirac equation: spin and angular momentum, nonrelativistic limit, antiparticles, chirality, Dirac equation + electromagnetic field

Alternative reading: Tuominen Sec. 9.1-9.2

Preliminary exercises, do these after reading the assignment work before the class of Wed 29th April and be prepared to present your solutions in class. Use this page to return the solutions.

Consider the (1+3)-dimensional Dirac Hamiltonian $H_D = c \vec \alpha \cdot \vec p + \beta m c^2,$ where $\vec \alpha = \gamma^0 \vec \gamma$ and $\beta = \gamma^0$ , and $\gamma^\mu$ satisfy the Clifford algebra. Define the orbital angular momentum operator $\hat {\vec L} = (\hat {\vec x} \times \hat {\vec p}) {\bf 1}$ and the operator for spin pointing in direction via $\hat \Sigma^i = \sigma^{jk} = i \gamma^j \gamma^k$ , where are in cyclic order (i.e., for , and etc.). Find the commutators $[H_D,\hat{ \vec L}]$ and $[H_D,\hat \Sigma^i]$ . You can do this in some specific representation of $\gamma^\mu$ or in a representation independent way.
Find the Dirac-Pauli and Weyl representations of the operator $\gamma_5=i\gamma_0\gamma_1\gamma_2\gamma_3$ . Define $L/R=(1\pm \gamma_5)/2$ and show that these are projection operators, i.e., that they satisfy , and .
In the presence of the electromagnetic field, the Dirac equation reads $[\gamma^\mu (i\hbar \partial_\mu-q A_\mu) -mc] \psi = 0,$ where $A^\mu = \begin{pmatrix} \frac{\varphi}{c} & \vec A\end{pmatrix}$ describes the scalar and vector potentials. Show that in the nonrelativistic limit this reduces to the Pauli equation.

Sixth week: relativistic quantum mechanics

Reading assignment for Tuesday April 21st, for class at noon:

Relativistic formulation: 4-vectors, Lorentz transformations, Lorentz invariance, Klein-Gordon equation, Dirac equation and its spinor solutions.

Alternative reading: Tuominen Sec. 8.1-8.2

Preliminary exercises, do these after reading the assignment work before the class of Tue 21st April and be prepared to present your solutions in class. Use this page to return the solutions.

Let us denote the transformation between contravariant and covariant vectors $A^\mu$ and $A_\mu$ by $A_\mu = g_{\mu \nu} A^\nu$ , where $g_{\mu\nu}$ is the metric tensor. Denote also the derivative with respect to contra/covariant vectors via $\partial_\mu \equiv \frac{\partial}{\partial x^\mu}$ and $\partial^\mu \equiv \frac{\partial}{\partial x_\mu}$ . Then find two examples of Lorentz invariant derivative forms containing second derivatives (of either scalars or four-vectors). Using the Minkowski metric, express them also in component form.
Assume a Lorentz invariant form of a dynamical equation for the wave function $\psi$ , of the form $i\hbar \frac{\partial}{\partial t} \psi(t,\vec r) = (-i\hbar c \vec \alpha \cdot \nabla + \beta mc^2) \psi(t,\vec r),$ where $\vec {\alpha}=\begin{pmatrix} \alpha^1 & \alpha^2 & \alpha^3 \end{pmatrix}$ and $\beta$ do not necessarily commute between each other. Requiring that $\psi(t,\vec r)$ also satisfies the Klein-Gordon equation, find the constraints (algebra) for $\alpha^j$ and $\beta$ .
Dirac equation can be represented by ( $\hbar=c=1$ ) $(i \gamma^\mu \partial_\mu - m) \psi = 0,$ where $\gamma^\mu = \begin{pmatrix} \gamma^0 & \vec \gamma \end{pmatrix}$ satisfy the Clifford algebra $\{\gamma^\mu,\gamma^\nu\} = 2 g^{\mu \nu} {\bf 1},$ and ${\bf 1}$ is the unit matrix. Considering both the (1+2) and (1+3) (time + space) dimensional cases,
1. Find example sets $\{\gamma^\mu\}$ ( $\mu=0,1,2$ or $\mu=0,1,2,3$ ) of smallest possible matrices satisfying the Clifford algebra.
2. Find the plane wave eigenfunctions of the Dirac equation corresponding to the representations chosen in (a) for .

Fifth week: quantum electrodynamics

Reading assignment for Tuesday April 14th, for class at noon:

Quantized electromagnetic field: field quantization in terms of bosonic operators, number states, coherent states, squeezed states, interaction of quantized radiation with atoms: coupling Hamiltonian, spontaneous emission, avoided level crossing

Mostly week 5 text, some material also in Tuominen Ch. 10, [Sakurai] Sec. 7.6

Preliminary exercises, do these after reading the assignment work before the class of Tue 14th April and be prepared to present your solutions in class. Use this page to return the solutions.

Express the electric field in terms of the creation and annihilation operators of photons.
Express the coherent state $|\alpha\rangle$ in terms of the Fock states.
Light-matter coupling:
1. Consider the Hamiltonian $H=\hbar \omega_c b^\dagger b + \frac{\hbar \omega_a}{2} \sigma_z + \hbar (g b^\dagger \sigma_- + g^* b \sigma_+)$ describing the coupling of the quantized electromagnetic field (operators $b, b^\dagger$ ) with an atomic excitation (Pauli matrices). Show that the total number of excitations in the system is conserved.
2. Avoided level crossing: Consider the Hamiltonian describing coupling of two systems, of the form $H=\begin{pmatrix} \epsilon_0/2 & g \\ g & -\epsilon_0/2 \end{pmatrix}.$ Plot the excitation spectrum (the two eigenvalues) as a function of $x=\epsilon_0/g$ . Identify the nature of the eigenstates in the three limiting cases $-x \gg 1$ , and $x \gg 1$ .

Fourth week: examples of interacting mean field theories

Reading assignment for Tuesday 31st March, for class at noon:
- Stoner and BCS models (Week 4 items)
Preliminary exercises, do these after reading the assignment work before the class of Tue Mar 31st and be prepared to present your solutions in class. Use this page to return the solutions.

Consider the case where the total energy of the system depends on some complex parameter $\psi$ in a form $E=E_0 + a |\psi|^2 + b |\psi|^4.$ with $a,b \in {\mathbb R}$ and
1. Find the range of parameters corresponding to the minimum taking place at (i) $|\psi|=0$ or (ii) $0<|\psi|<\infty$ .
2. Sketch vs. $\psi$ in those two cases. Hint: you can write $\psi=|\psi|e^{i\phi}$ , $\phi \in {\mathbb R}$ .
Consider the Stoner model (spin polarized Fermi gas) for the 2d case. Derive the single-particle, Hartree, and Fock energy terms in terms of the spin polarization , and find the regime of parameters where the energy minimum corresponds to a magnetic state ( $P \neq 0$ ).
Consider a Hamiltonian of the form $H=\sum_{\sigma=\uparrow/\downarrow} \epsilon_0 c_\sigma^\dagger c_\sigma + \delta c_\uparrow^\dagger c_\downarrow^\dagger + \delta^* c_\downarrow c_\uparrow,$ where $\delta$ is a complex number and $c_\sigma$ annihilates an electron with spin $\sigma$ (we hence here ignore all other quantum numbers of the single-electron state besides spin). Try to find new fermionic operators $d_\sigma$ that allow presenting this in terms of a regular "free-fermion" form $H=\sum_\sigma \Omega d_\sigma^\dagger d_\sigma + {\rm const.}$ , where $\Omega$ depends on $\epsilon_0$ and $\delta$ . Hint: Bogoliubov transformation.

Third week: many body quantum mechanics continued

Reading assignment for Tuesday 24th March, for class at noon:
- One- and two-particle Fock-space operators
- Non-interacting Fermi gas and its properties
- Electron-hole formalism
- Hartree-Fock (mean field) method

You can also read this in Tuominen, Ch. 7.

Preliminary exercises, do these after reading the assignment work before the class of Tue Mar 24th and be prepared to present your solutions in class. Use this page to return the solutions.
1. Write the free-fermion Hamiltonian (nonrelativistic kinetic energy) in terms of the Fock-space operators in
  1. In terms of arbitrary single-particle basis states $\{|\mu\rangle\}$
  2. Momentum representation
  3. Position representation
2. Find the ground-state energy density in terms of the particle density in a two-dimensional Fermi gas.
3. The Fermi gas is a state of the form $|F\rangle = \prod_{\sigma,|{\bf p}|\le p_F} a_{{\bf p} \sigma}^\dagger |0\rangle,$ where $a_{{\bf p}\sigma}^\dagger$ create fermions with momentum ${\bf p}$ and spin $\sigma$ , and $|0\rangle$ is a vacuum state. However, $|F\rangle$ can also be treated as a vacuum of a set of operators. Find those operators.

Should there be product in the Fermi gas state?

— 23 Mar 20

Yes, thanks. Corrected.

— 23 Mar 20

I noticed that the order in 1st exercise was a bit strange, so I changed it. It does not matter if you did it in another order.

— 24 Mar 20

Second week: start of many-body quantum mechanics

Reading assignment for Tuesday 17th March, for class at noon:
- Identical particles and their permutation symmetry
- Bosons and fermions, their many-particle wave functions
- Fock space: states and creation/annihilation operators
- Field operators
- Wick's theorem
You can read this from [Sakurai] Ch. 7, or Tuominen Ch. 7. An interesting read is also [Nazarov] Ch. 2-3.

Preliminary exercises, do these after reading the assignment work before the class of Tue Mar 17th and be prepared to present your solutions in class. Use this page to return the solutions.
1. Consider three single-particle states with wave functions $\phi_i(\vec x,s_z)$ and energies $\epsilon_i$ (). Find all possible three-particle (a) bosonic and (b) fermionic wavefunctions along with their energies. Express them also in occupation number representation. Hint: there may be many of them, but it is enough to show the different types of wavefunctions.
2. Consider the fermionic creation and annihilation operators and $a_i^\dagger$ and the bosonic creation and annihilation operators and $b_i^\dagger$ with $i=1,2,\dots$ indexing the corresponding single-particle state.
  1. Express the fermionic occupation-number state $|101\rangle$ in terms of the creation operators and the vacuum state $|000\rangle$ . Here the first item refers to , the second to , and so on, and the states with are assumed to be unoccupied.
  2. Express the bosonic occupation-number state $|201\rangle$ in terms of the creation operators and the vacuum state $|000\rangle$ . Here the first item refers to , the second to , and so on, and the states with are assumed to be unoccupied.
  3. Calculate the commutators $[a_\nu^\dagger a_\nu,a_{\mu}^\dagger a_{\mu}] \text{ and } [b_\nu^\dagger b_\nu,b_{\mu}^\dagger b_{\mu}].$
3. Using Wick's theorem, calculate the following matrix elements
  1. $\langle 11| a_j^\dagger a_j |11\rangle$
  2. $\langle 11| a_j^\dagger a_j^\dagger |11\rangle$
  3. $\langle 11| a_k^\dagger a_j^\dagger a_k a_j |11\rangle$
  Here $j,k\in \{1,2\}$ . Consider both fermionic and bosonic operators .

First week: path integrals

Reading assignment for Tuesday 10th March:

Path-integral formulation of the problem of one particle on a 1d potential, and the free-particle path integral
Alternative: Tuominen, Sec. 6.1-6.4

Preliminary exercises, do these after reading the assignment work before the class of Tue Mar 10th and be prepared to present your solutions in class.
1. With a (single-particle) Hamiltonian $\hat H=\frac{\hat p^2}{2m} + \hat V$ , where $\langle x | \hat V | x'\rangle=V(x) \delta(x-x')$ , find an estimate for the infinitesimal time propagator $\langle x | e^{-\frac{i}{\hbar} \hat H \delta t} |x'\rangle$ in the limit $\delta t \rightarrow 0$ (including the first-order term of $\delta t$ in the exponent).
2. Using the classical action $S=\int_{t_1}^{t_2} dt L(x,\dot x,t),$ $L(x,\dot x,t)=\frac{m \dot x^2}{2} - V(x)$ , find the constraint for the path that minimizes the action.
3. Denote the vector $v_N^T=\begin{pmatrix} x_1 & x_2 & \cdots & x_N \end{pmatrix}$ . Calculate the Gaussian integral $\prod_{j=1}^N \int d x_j \exp\left[-v^T A v\right],$ where is an invertible $N\times N$ matrix whose eigenvalues have a positive semidefinite real part.