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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