課程名稱︰物理化學二─量子力學 課程性質︰化學系大二必修 課程教師︰鄭原忠 開課學院：理學院 開課系所︰化學系 考試日期（年月日）︰3/29/2011 考試時限（分鐘）：120min 是否需發放獎勵金：是 試題 : Physical Chemistry II: Quantum Chemistry Mid-term Exam #1 Date:3/29/2011 ***Refer to the last page for formulas of frequently used intergrals*** 1(20%)A particle with mass m is in a one-dimensional box defined by the potential energy function : ∞ if x<or=0 V(x)={ 0 if 0<x<a ∞ if x>or=a Consider that the particle is described by the wave functionψ=Nx(a-x). (a)Calculate the normalization constant N (b)What is the probability of finding the particle in the middle third of the box? (c)What is the average energy of the particle? (d)Now considering a meassurement in the energy ground state. Note that the ground-state wavefunction of the 1-D particle in-a-box problem is 1/2 φ=(2/a) sin(πx/a) ^ *^ ^ * 2(10%)A Hermitian operator A satisfies ∫ψAφ dτ=∫φ(Aψ)dτ for arbitrary wavefunctions ψ and φ. Show that in 1-D, the kinetic energy ^ ^ 2 ^ operator T=Px /2m is Hermitian (hint: Px=-ih d/dx)←這個h是h上面加一槓 3.A quantum particle with mass m in a harmonic potential is described by the ^ ^2 2 2 Hamiltonian H=P/2m + mωx /2 .Define the non-Hermitian ladder operators as ______ ^ ^ + ______ ^ ^ a=√mω/2h (x+iP/mω) a = √mω/2h (x-iP/mω) (h還是要加一槓) ^ + ^ + We have showed that H can be re-written in terms of a and a as H=hω(aa+1/2) _ + ___ (h加一槓) In addition, aψn(x) = √n ψn-1(x) and aψn(x)=√n+1 ψn+1(x) ^ where ψn denotes the eigenfunctions of H with vibrational quantum number n= 0,1,2..... Answer the following questions using the properties of the ladder operators: ∞ * ^ (a)(5%) Evaluate En= ∫ ψn(x)Hψn(x) dx using ladder operators to find the energy levels En -∞ 1/4 -αx^2 (b)(5%) The ground state wavefunction is ψ0(x)=(α/π) e with α=mω/h What is the wavefunction of the first excited state ψ1(x)? + Hint: consider aψ0(x). (c)(5%) Consider a state that is described by the wavefunction ψ(x)=c[ψ1(x) + ψ2(x)]. What is the value of the normalization constant c (assuming a real number)? What is the avaerage energy and the standard deviation in energy? ^ ^ (d)(10%) Calculate the expectation value <x> and <x^2> for the state ψ (e)(5%) Sketch rough graphs of ψ1(x), ψ2(x) and ψ in the harmonic potential. Label the energy levels. Use the interference of waves to explain what you found in (d) -αx^2 4(10%) Show that the function e satisfied the Schrodinger equation for a quantum harmonic oscillator, What conditions does this place on α? What is the average energy E of this state? ^ 5(a)(5%) Prove that if A is a Hermitian operator, then the expectation value of ^2 ^2 A for any wavefunctions must be greater or equal to zero i.e <A> >or=0 (b)(5%) Use the above statement to explain why ta harmonic oscillator whose energy expectation value equals to zero must violate the Heisenberg`s uncertainty principle. 6 Answer true or false for the following statements(3 points each) (a)For a 1-D quantum system described by the wavefunction ψ(x), the ^ expectation value of an observables A is calculated using the intergral: ∞ ^ 2 ∫ A|ψ(x)|dx. -∞ (b)Solving the time-independent Schrodinger equation is needed in order to calculate the energy of a quantum state. (c)The zero point energy is lower for a Helium atom in a box than for an electron (d)According to the superposition princicle, if g1(x) and g2(x) are both ^ eigenfunctions of the linear operator A, then their linear combinations are also ^ eigenfunctions of A (e)The wavefunction of a system must satisfy the time-independent Schrodinger equation (f)Observable quantities must be eigenvalues of quantum-mechanical operators (g)For the n=9 harmonic oscillator energy eigenfunction, the sign of ψ in the right-hand classical forbidden region is opposite the sign of ψ in the left-hand classical forbidden region. 7.Bonus questions(5 points each) (a)Consider the system described in Problem1. In the box, the Hamiltonian ^ ^ commutes with the momentum operator, i.e [H,P]=0. Explain why they do not share the same eigenbasis. (b)Electron tunneling occurs in the scanning tunneling microscope(STM), which makes possible atomic resolution of surfaces. Explain why? Hint:use the distance dependence of electronic tunneling probabilities --

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