課程名稱︰量子力學二
課程性質︰必修
課程教師︰高涌泉
開課學院：理學院
開課系所︰物研所
考試日期（年月日）︰2013.04.25
考試時限（分鐘）：110分
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Consider a 1-dimensional simple harmonic oscillator. The Hamiltonian is
2
p 1 2 2
H= ---- + ---mω x . With the help of the lowering operator a and the
2m 2
﹢ √mω ﹢√mω
raising operator a (a=-----*(x+ip/mω),a =-----*(x-ip/mω)), evaluate
√2h √2h
2 2
<n1|x |n2> and <n1|p |n2> in which |ni> is the eigenstate of H with
eigenvalue E_ni=hω(ni+1/2).
(h 為 h bar)
2. Consider a charged particle bound in the harmonic oscillator potential
1 2 2
V(x)=---mω x . A weak electric field ε is applied to the system such
2
that the potential energy is shifted by an amount H'=-qεx. Calculate the
energy levels of the perturbed system to 2nd order in the small
perturbation.
3. Give the definition of the magnetic flux quantum (or quantum of magnetic
flux) and explain its meaning.
4. Show that the transition amplitude <xf,tf|xi,ti> can be expressed as a path
integral N∫D[x(t)]exp(iS[x(t)]/h).
(h 為 h bar)
5. How did Einstein argue for the existence of the spontaneous emission
process?
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