Download Quantum Mechanics

HOWARD UNIVERSITY
WASHINGTON, DC, 20059
DEPARTMENT OF PHYSICS AND ASTRONOMY
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—Quantum Mechanics—
1. Consider a simple particle moving in a one-dimension and under the influence of a
restoring force F = −kx5 , k > 0.
a. Write down the Schrödinger equation for the wave-function Ψ(x) of such a particle.
b. Show that for very large x, Ψ(x) decays exponentially with some power of x; find this
asymptotic wave-function Ψasymp (x) = φ(x).
c. Writing Ψ(x) = φ(x)f (x), find the differential equation that determines f (x).
d. Consider now varying k and independently n for the restoring force F = −kxn . Which
of these two variations does (does not) preserve the qualitative features of the energy
spectrum, and why (why not).
2. The hydrogen atom wave function may be written as R(r)Y`m (θ, φ), where R is the
radial function and Y`m are the spherical harmonics.
a. What is the differential equation for R(r)?
b. The differential equation may be simplified somewhat by changing r and E into ρ =
r/a0 and W = E/[ke2 /(2a0 )], where a0 = h̄2 /(mke2 ) is the Bohr radius, k is the
Coulomb law constant, e is the electron charge and m is the electron’s mass. What is
this simplified differential equation for R(r).
c. Show that R(r) approaches ρ` for small ρ.
d. For the ground state, we may try R = ρ` e−αρ . Calculate then α and W .
In spherical coordinates,
∇2 ψ =
∂ 2 ψ 2 ∂ψ
1
∂ ¡
∂ψ ¢
1
∂2ψ
+
.
+
sin
θ
+
∂r2
r ∂r
r2 sin θ ∂θ
∂θ
r2 sin2 θ ∂ϕ2
1
3. The n = 2 level of the hydrogen atom is fourfold degenerate (neglecting spin), where
the normalized whave functions are:
r
³
r
1
1
r´ − r
2
Ψ200 = √
e 2a ,
Ψ21m =
re− 2a Y1m ,
2−
5
a
4 3a
4 2πa3
r
Y10 =
r
3
cos θ ,
4π
Y1,±1 =
3
sin θ e±iϕ ,
8π
~ along the z-axis and apply the
where a is the Bohr radius. Apply now an electric field E
degenerate perturbation theory to first order.
a. What is the perturbation Hamiltonian H 0 ?
b. What is the matrix (secular) equation for H 0 ? Show that there are only two non-zero
matrix elements.
c. What are the eigenvectors, eigenvalues and degeneracies after matrix diagonalization?
R∞
(A possibly useful integral: 0 dx xn e−x = n!.)
4. A particle moves in one dimension and in a potential of the form V (x) = 0, for |x| < a
and V (x) = V0 > 0 for |x| > a. The particle has energy 0 < E < V0 .
a. Solve the Schrödinger equation in each of the three regions: I: −∞ < x < −a,
II: −a < x < +a and III: +a < x < +∞.
b. Specify the continuity conditions at x = −a and x = +a, and classify the solutions
by their parity (behavior under the x → −x reflection).
c. Show that the continuity conditions restrict the possible values of energy E and find
the equation for E.
d. Show that there is always at least one bound state. Is it (anti)symmetric with respect
to the x → −x reflection?
e. For a → 0, but such that aV0 = const., calculate the energy level of the bound state.
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