[題目]量子力學一問
[領域] (題目相關領域)
[來源] (課本習題、考古題、參考書...)
[題目]
(a)Consider a particle governed by H = (-h^2/2m)d^2/dx^2+V(x):
Now, if the particle is further confined to be at x = na with n be integers
and a being a positive constant,
one needs to discretize the system:
replacing any function f(x = na) by fn and d^2/dx^2Ψ(x)∣x=na
by((Ψn+1) + (Ψn-1) +2Ψn)/a^2 .
Let us denote the state when the particle is localized at x = na by ∣n＞
and assume that ＜n∣m＞=δnm.
Express H in terms of the Dirac notations (ket and braand etc.)
(b) In the above discrete system, the momentum operator should be
changed accordingly. However, there are many possible ways to do:
For example,(h/i)(dΨ(x)/dx)=(h/i)((Ψn+1)-Ψn)/a or
(h/i)(dΨ(x)/dx)=(h/i)((Ψn+1)-(Ψn-1))/2a
which one is correct? why?
(c)5 Suppose now that the particle is in the state
∣Φ＞≡∣1＞+i∣2＞+2∣3＞.
Evaluate the averaged value of energy for ∣Φ＞.
(d)Now suppose that the particle is further confined to be at
n = 1 and n = 2, i.e., Ψn = 0
when n = 3; 4; 5...... and n = 0,-1,-2....
The wavefunction ∣Φ(t)＞of the particle at time
t can be thus written as ∣Φ(t)＞=α(t)∣1＞+β(t)∣2＞
Find the differential equations that
α(t) and β(t) obey (5).
If at t = 0, ∣Φ(0)＞=∣1＞,find∣Φ(t)＞at t>0.(5)
[瓶頸] (寫寫自己的想法，方便大家為你解答)
a小題希望我將H算符換成狄拉克符號，但我根本不知道要怎樣換= =
所以完全無頭緒
b小題則是希望我驗證兩個式子哪個是對的
但我做出來都和a小題牴觸
不曉得原因在哪= =
c.d則是連題目都看不太懂
希望版上高手能幫小弟指點迷津
非常感謝
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