Problems, set 8. PHYSICS 212A: Mathematical Physics
1. The one-dimensional Schr¨odinger’s equation (SE) for a particle of mass mis:
−¯h2
2m
d2ψ
dx2+ (V(x)−E)ψ= 0
¯his the Plank’s constant, V(x) is the potential, Eis the energy. Obtain a WKB wave-like
solution (x > a,E > V (x), V(a) = E) in the form ΨW K B =f·exp(g) using the ansatz:
ψ= exp ·1
¯hZ(ϕ0+ ¯hϕ1+. . .)dx¸,with the unknown ϕ0,ϕ1, ...
Treat ¯has a parameter, and collect terms with different powers of ¯hin the SE.
[Hint. You will need only two terms in the exponent for the WKB approximation.]
(a) First, obtain ΨW KB with the prefactor fand exponential function g, expressed throught
V,E, etc., for arbitrary V(x) and E.
(b) Expand V(x) near the stopping point x=aand use the linearized form of V(x) to
obtain a more standard form of ΨW KB .
2. Solve the differential equation for the Green’s function using Fourier transform:
"d2
dx2−a2#G(x, x0) = δ(x−x0)
−∞ < x < ∞. Boundary conditions are: G(±∞, x0) = 0.
3. M&W: 9-7. Answer should be compact (and real).
[Hint: Apply BCs, use the “stitching” technique.]
4. Find the Green’s function GE(x, x0) of the 1D Schr¨odinger’s equation for a free particle of
mass musing the “stitching” technique:
³ˆ
H − E´GE=−¯h2
2m
∂2
∂x2GE−EGE=δ(x−x0)
for E < 0, such that GE→0 as |x−x0| → ∞.
5. (a) Using the GEobtained in the Problem #4, derive the integral equation on the wave
function ΨEfor a particle in the potential U(x) (U(x)→0 when |x|→∞). This will
be the integral form of the Schr¨odinger’s equation, valid for E < 0.
[Hint. Consider f(x) = −U(x)ΨE(x)as the “source” term.]
(b) Using the integral form of the Schr¨odinger’s equation find the normalized wave-function
and the energy of the bound state in the “δ-functional well”: U(x) = −αδ(x).
6. M&W: 4-13. Use the inverse Laplace transform with contour integration explicitly (no tables
of LT and/or convolution theorem).
[Hint. Make sure that the LT image-function, F(s), is analytic on the RHside of the contour.
7. M&W: 4-14. Use contour integration.
8. Solve, using the Laplace transform:
y00 +a2y=e−x
with a=real and y0(0) = y(0) = 0. Use contour integration for the inverse LT (no tables).