[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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