#time evolution

Dirac delta function potential

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Q:Dirac delta function potential

Problem 2.24/2.25 [1] introduces a Dirac delta function potential

\begin{equation}\label{eqn:diracPotential:20} H = \frac{p^2}{2m} – V_0 \delta(x), \end{equation}

which vanishes after \( t = 0 \). Solve for the bound state for \( t < 0 \) and then the time evolution after that.

A:

The first part of this…

Two spin time evolution

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Motivation

Our midterm posed a (low mark “quick question”) that I didn’t complete (or at least not properly). This shouldn’t have been a difficult question, but I spend way too much time on it, costing me time that I needed for other questions.

It turns out that there isn’t anything fancy required for this question, just perseverance and…

Time evolution of spin half probability and dispersion

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Question: Time evolution of spin half probability and dispersion ([1] pr. 2.3)

A spin \( 1/2 \) system \( \BS \cdot \ncap \), with \( \ncap = \sin \beta \xcap + \cos\beta \zcap \), is in state with eigenvalue \( \Hbar/2 \), acted on by a magnetic field of strength \( B \) in the \( +z \) direction.

(a)

If \( S_x \) is measured at time \( t…

PHY1520H Graduate Quantum Mechanics. Lecture 2: Basic concepts, time evolution, and density operators. 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 chap 1 (basic concepts),3 (density operator). content from [1].

Basic concepts

We’ve reviewed the basic concepts that we will encounter in Quantum…

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Question: Quantum Virial Theorem ([1] pr. 2.7)

Consider a particle with Hamiltonian

\begin{equation}\label{eqn:qmVirialTheorem:20} H = \frac{\Bp^2}{2 m} + V(\Bx), \end{equation}

By calculating the time evolution of \( \antisymmetric{\Bx \cdot \Bp}{H} \), identify the quantum virial theorem and show the conditions where it is satisfied.

Answ…

A symmetric real Hamiltonian

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Question: A symmetric real Hamiltonian ([1] pr. 2.9)

Find the time evolution for the state \( \ket{a’} \) for a Hamiltian of the form

\begin{equation}\label{eqn:symmetricHamiltonianEvolution:20} H = \delta \lr{ \ket{a’}\bra{a’} + \ket{a”}\bra{a”} } \end{equation}

Answer

This Hamiltonian has the matrix representation

\begin{equation}\label{…

Heisenberg picture position commutator

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Question: Heisenberg picture position commutator ([1] pr. 2.5)

Evaluate

\begin{equation}\label{eqn:positionCommutator:20} \antisymmetric{x(t)}{x(0)}, \end{equation}

for a Heisenberg picture operator \( x(t) \) for a free particle.

Answer

The free particle Hamiltonian is

\begin{equation}\label{eqn:positionCommutator:40} H = \frac{p^2}{2m},

Dynamics of non-Hermitian Hamiltonian

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Question: Dynamics of non-Hermitian Hamiltonian ([1] pr. 2.2)

Revisiting an earlier Hamiltonian, but assuming it was entered incorrectly as

\begin{equation}\label{eqn:dynamicsNonHermitian:20} H = H_{11} \ket{1}\bra{1} + H_{22} \ket{2}\bra{2} + H_{12} \ket{1}\bra{2}. \end{equation}

What principle is now violated? Illustrate your point…