Page 137 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

128 Harmonic oscillator
3
ˆ (n x ‡ n y ‡ n z ‡ )"ù ˆ (n ‡ )"ù (4:56)
3
E n x ,n y ,n z
2 2
where n is called the total quantum number. All the energy levels for the
isotropic three-dimensional harmonic oscillator, except for the lowest level, are
degenerate. The degeneracy of the energy level E n is (n ‡ 1)(n ‡ 2)=2.

Problems

4.1 Consider a classical particle of mass m in a parabolic potential well. At time t the
displacement x of the particle from the origin is given by
x ˆ a sin(ùt ‡ b)
where a is a constant and ù is the angular frequency of the vibration. From this
expression ®nd the kinetic and potential energies as functions of time and show
that the total energy remains constant throughout the motion.
4.2 Evaluate the constant c in equation (4.10). (To evaluate the integral, let y ˆ
cos è.)
4.3 Show that ^ a and ^ a in equations (4.18) are not hermitian and that ^ a is the adjoint
y
y
of ^ a.
^
4.4 The operator N ^ a ^ a is hermitian. Is the operator ^ a^ a hermitian?
y
y
^
^
4.5 Evaluate the commutators [H, ^ a] and [H, ^ a ].
y
6
4.6 Calculate the expectation value of x for the harmonic oscillator in the n ˆ 1
state.
4.7 Consider a particle of mass m in a parabolic potential well. Calculate the
probability of ®nding the particle in the classically allowed region when the
particle is in its ground state.
4.8 Consider a particle of mass m in a one-dimensional potential well such that
2 2
1
V(x) ˆ mù x , x > 0
2
ˆ1, x , 0
What are the eigenfunctions and eigenvalues?
4.9 What is the probability density as a function of the momentum p of an oscillating
particle in its ground state in a parabolic potential well? (First ®nd the
momentum-space wave function.)
4.10 Show that the wave functions A n (ã) in momentum space corresponding to
ö n (î) in equation (4.40) for a linear harmonic oscillator are

1
A n ( ã) ˆ (2ð) ÿ1=2 ö n (î)e ÿi ãî dî
ÿ1
2
n
ˆ i ÿn (2 n!ð 1=2 ÿ1=2 ÿ ã =2 H n (ã)
)
e
where î (mù=") 1=2 x and ã (m"ù) ÿ1=2 p. (Use the generating function (D.1)
to evaluate the Fourier integral.)