課程名稱︰量子力學一
課程性質︰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）︰108年10月23日
考試時限（分鐘）：unknown
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
註：為方便表示數學符號，部分符號以LaTeX碼輸入。
如：a^{b}_{c}表a上標b下標c；\vec{v}表v向量；\hat{x}表x單位向量；\cdot表向量內積
10/23/2019
PHYS7014 Quantum Mechanics I
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 deri-
vedresults in the main text of Sakurai and Napolitano up to the range of the
exam (i.e., Chapter 1) 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 questions about the problems or notations. In your answers, define
clearly your notations if they differ from those in the main textbook.
Problems
Consider a spin-1/2 particle placed in a uniform magnetic field,
\vec{B}=\frac{B_0}{\sqrt{2}}(\hat{x}+\hat{z}),
where B_0 >0. The dynamics of the particles are governed by the Hamiltonian
H=-γ\vec{S}\cdot\vec{B},
where \vec{S} is the vector spin operator for the particle, and γ is a positi-
ve constant.
1.[15]In the original Stern-Gerlach experiment, a beam of silver atoms (each
carries 47 electrons) goes through a magnetic field
\vec{B}=B_0\hat{z},
where B_0 has a spatial dependence, and split into two.
(a)Explain why B_0 cannot be a constant.
(b)Explain why they can be seen as a spin-1/2 particles.
(c)Explain why the nuclear spin can be safely ignored in this case.
2.[5]Calculate the matrixof the Hamiltonian in the {|+>,|->} basis with
S_Z|±>=±\hbar/2|±>.
3.[10]Calculate the eigenvalues and the normalized eigenvectors of H. For the
normalized eigenvectors, write them in the form of N(1 ), where N and α
α
are numbers.
4.[5]What is the matrix that diagonalizes H, with the elements of the diagon-
alized H listed in the ascending order?
5.[10]At time t=0, the system is in the state |->, and the energy is measured.
What are the possible outcomes of the energy measurement and with what prob-
abilities?
6.[5]In the previous problem, what is the expectation value of the energy mea-
surement?
7.[10]Evaluate Δs_xΔs_z for the lower energy eigenstate, where Δs_x and
Δs_z denote the uncertainties of S_x and S_z operators, respectively.
8.[5]Explain whether your result in the previous problem violates the uncerta-
inty relation.
9.[10]Suppose the spin-1/2 particle is initially in the spin-up state along
the z axis, and has a fixed momentum \vec{p}. Define a ket vector to descri-
be such a state. Derive the final ket vector after one performs a finite tr-
anslation of Δ\vec{x} to the particle. Explain what happens to the spin st-
ate.
10.[10]Suppose \hat{n} is a unit vector in the (θ,φ) direction, as shown in
Fig.1(a). Find the normalized |ψ> such that
(\vec{S}\cdot\hat{n})|ψ>=+\hbar/2|ψ>. (4)
Express your result in the {|+>,|->} basis. Take the convention that the
coefficient of |+> is real.
11.[10]any spin-1/2 state can be visualized in the three dimensional space,
called the Bloch sphere as shown in Fig. 1(b), as a Bloch vector, (<σ_x>,
<σ_y>,<σ_z>), evaluated with respect to the given state. Compute the
Bloch vector for |ψ> in the previous problem, and show that it falls on
the surface of the Bloch sphere. (The last property is a characteristic of
pure states.)
12.[5]In the textbook, the spin raising and lower operators of the spin-1/2
particle are introduced respectively as
S_+ ≡\hbar|+><-|, S_- ≡\hbar|-><+|. (5)
Suppose we know that S_±=S_x±iS_y even for higher spin systems. Explain
whether one can assume for a spin-s particle that the raising and lowering
operators are
S_+ =N\hbar[|s,s><s,s-1|+|s,s-1><s,s-2|+...+|s,-s+1><s,-s|], (6)
S_- =N\hbar[|s,s-1><s,s|+|s,s-2><s,s-1|+...+|s,-s><s,-s+1|],
where |s,m> denotes a spin-s state with s_z =m, and N is some overall fact-
or.