Hamiltonian for a scalar field

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In [1] it is left as an exersize to expand the scalar field Hamiltonian in terms of the raising and lowering operators. Let’s do that.

The field operator expanded in terms of the raising and lowering operators is

\begin{equation}\label{eqn:scalarFieldHamiltonian:20} \phi(x) = \int \frac{ d^3 k}{ (2 \pi)^{3/2} \sqrt{ 2 \omega_k } } \lr{ a_k…

Spin three halves spin interaction

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Q: [1] pr 3.33

A spin \( 3/2 \) nucleus subjected to an external electric field has an interaction Hamiltonian of the form

\begin{equation}\label{eqn:spinThreeHalvesNucleus:20} H = \frac{e Q}{2 s(s-1) \Hbar^2} \lr{ \lr{\PDSq{x}{\phi}}_0 S_x^2 +\lr{\PDSq{y}{\phi}}_0 S_y^2 +\lr{\PDSq{z}{\phi}}_0 S_z^2 }. \end{equation}

Show that the…

PHY1520H Graduate Quantum Mechanics. Lecture 13: Time reversal (cont.), and angular momentum. Taught by Prof. Arun Paramekanti

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Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

These are notes for the UofT course PHY1520, Graduate Quantum Mechanics, taught by Prof. Paramekanti, covering \textchapref{{4}}, \textchapref{{3}} [1] content.

Time reversal (cont.)

Given a time reversed state

\begin{equation}\label{eqn:qmLecture13:20} \ket{\…

Expectations for SHO Hamiltonian, and virial theorem.

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Question: Expectations for SHO Hamiltonian, and virial theorem. ([1] pr. 2.3) (a)

For a 1D SHO, compute \( \bra{m} x \ket{n}, \bra{m} x^2 \ket{n}, \bra{m} p \ket{n}, \bra{m} p^2 \ket{n} \) and \( \bra{m} \symmetric{x}{p} \ket{n} \).

(b)

Verify the virial theorem is satisfied for energy eigenstates.

Answer (a)

Using

\begin{equation}\label{e…

PHY1520H Graduate Quantum Mechanics. Lecture 4: Quantum Harmonic oscillator and coherent states. Taught by Prof. Arun Paramekanti

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Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough. This lecture reviewed a lot of quantum harmonic oscillator theory, and wouldn’t make sense without having seen raising and lowering operators (ladder operators), number operators, and the like.

These are notes for the UofT course PHY1520, Graduate Quantum…