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Dirac delta function potential
Dirac delta functionĀ 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} ā 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 thisā¦
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Two spin time evolution
Two spin timeĀ evolution
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Motivation
Our midterm posed a (low mark āquick questionā) that I didnāt complete (or at least not properly). This shouldnāt 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 isnāt anything fancy required for this question, just perseverance andā¦
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Time evolution of spin half probability and dispersion
Time evolution of spin half probability andĀ 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 \( tā¦
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PHY1520H Graduate Quantum Mechanics. Lecture 2: Basic concepts, time evolution, and density operators. Taught by Prof. Arun Paramekanti
PHY1520H Graduate Quantum Mechanics. Lecture 2: Basic concepts, time evolution, and density operators. Taught by Prof. ArunĀ Paramekanti
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Disclaimer
Peeterās 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
Weāve reviewed the basic concepts that we will encounter in Quantumā¦
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Quantum Virial Theorem
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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.
Answā¦
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A symmetric real Hamiltonian
A symmetric realĀ Hamiltonian
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Question: A symmetric real Hamiltonian ([1] pr. 2.9)
Find the time evolution for the state \( \ket{aā} \) for a Hamiltian of the form
\begin{equation}\label{eqn:symmetricHamiltonianEvolution:20} H = \delta \lr{ \ket{aā}\bra{aā} + \ket{aā}\bra{aā} } \end{equation}
Answer
This Hamiltonian has the matrix representation
\begin{equation}\label{ā¦
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Heisenberg picture position commutator
Heisenberg picture positionĀ 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},
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Dynamics of non-Hermitian Hamiltonian
Dynamics of non-HermitianĀ 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 pointā¦
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