課程名稱︰量子力學一
課程性質︰物理系系定選修
課程教師︰蔣正偉
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2019年12月4日
考試時限（分鐘）：未知
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
註：為明確表示數學式，部分較難以圖像表示之部分以\LaTeX碼表示
ex. \vec{a} 表 a向量, a^{\dagger} 表a dagger, \sart{a}表根號a, \hbar 表 hbar,
\int^{b}_{a}表自a積分至b
12/04/2019
PHYS7014 Quantum Mechanics I ─ Second Midterm Exam
INSTRUCTIONS
This is an open-book, 120-minute exam. Your are only allowed to use the main
textbook (by Sakurai and Napolitano) and your own handwritten notes. Only
derived results in the main text of Sakurai and Napolitano up to the range of
this exam (i.e., Chapter 2) can be used. The score of each sub-problem is
indicated by the number in square brackets. To avoid any misunderstanding, ask
if you have any queations about the problems or notations. In your answers,
define clearly your notations if they differ from those in the main textbook.
Note that we also do not distinguish the notation of an operator from that of
its corresponding variable.
You may find some of the following formulas useful:
\int^{x}\sqrt{1-u^2}du=1/2(x\sqrt(1-x^2}+arcsin(x))+C
\int_{-∞}^{∞}exp(-(ax^2+bx+c))=(π/α)^(1/2)exp((b^2-4ac)/4a)) (a>0)
PROBLEMS
1.[70 points] Consider a one-dimensional simple harmocis oscillator (SHO) with
the potential given by
V(x)=1/2mω^2x^2.
The n-th energy level is denoted by |n>. Recall that
a } = 1/\sqrt{2}(x/x_0π±ix_0/\hbar p)
a^{\dagger} }
a } = {\sqrt{n}|n-1>
a^{\dagger} } {\sqrt{n+1}|n+1> ,
where x_0≡\sqrt{\hbar/(mω)}. Also, recall that the position and momentum
operators in the Heisenberg pircture for the SHO are solved to be
{x(t)=x(0)cos(ωt)+p(0)/(mω)sin(ωt),
{p(t)=-mωx(0)sin(ωt)+p(0)cos(ωt).
(a)[10] Work out the matrix forms of the following operators: a, a^{\dagger}
, x and p in the Schrodinger picture. Show at least the upper left 4×4
sub-matrix.
(b)[10] Calculate (Δx)(Δp) for the n-th level state, where for your
convenience Δa≡\sqrt{<A^2>-<A>^2}.
(c)[10] Calculate the expectation values of kinetic and potential energy of
the n-th level state, and show that they satisfy the virial theorem.
(d)[10] Compute the expectation values of x(t) and p(t) for the n-th level
state. Explain physically what your results mean. In particular, are they
consistent with the classical picture of an oscillator?
(e)[10] Use the WKB approximation method to work out the eigenstate energies.
Define the coherent state, denotes by |α>, satisfying
a|α>=α|α> with the normalization <α|α>=1
(f)[10] Derive the normalized coherent state in the basis of {|n>}.
(g)[10] Explain what kind of quantity the eigenvalue α can be, and whether
or not one can set it to be real at all times.
2.[10 points] Consider a one-dimensional quantum mechanical problem of a
particle with a time-independent Hamiltonian H. In the path integral
calculation, we have shown in class that for a particle with a general
Hamiltonian H(p,x):
<x_{i+1}|H|x_i>=\int^{∞}^{-∞}(dp/(2π\hbar))H(p,\bar{x}_i)×
exp(i/\hbar p(x_{x+1}-x_i))
where x_i and x_{i+1} denote respectively the positions of the particle at
two infinitesimally separated moments t_i and t_{i+1}. Explain why one should
use \bar{x}_i=(x_i+x_{i+1})/2 instead of x_i or x_{i+1}.
3.[20 points] Consider a particle of electrin charge Qe and mass m in an
electromagnetic potential, described by the Hamiltonian
H=Π^2/(2m)+Qeφ with Π=\vec{p}-Qe/c\vec{A},
where φ(\vec{x}) and \vec{A}(\vec{x}) are time-independent scalar and
vector potentials, respectibely. Use the Heisenberg pircture throughout this
problem.
(a)[10] Explain whether d\vec{x}/dt and \vec{x} can commute with each other.
If so, prove it. If not, give a simple example.
(b)[10] Derive the quantum mechanical version of the Lorentz force:
m(d^\vec{x}/dt^2)=dΠ/dt=Qe[\vec{E}+1/(2c)×
(d\vec{x}/dt×\vec{B}-\vec{B}×d\vec{x}/dt)].