Download Quantum Mechanics

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—Quantum Mechanics—
1. Develop a variational principle from the Schrödinger equation
−
h̄2 ~ 2
∇ ψλ + V (~r)ψλ = Eλ ψλ ,
2m
where ψλ is the wave-function of the stationary state with energy Eλ , and V (~r) is a nonnegative real potential.
R
a. Using that d3 ~rψλ∗ ψλ0 = δλ,λ0 , obtain an integral expression for Eλ , in terms of ψλ ,
~ λ , their conjugates and V (~r), but no higher derivatives. Show that Eλ ≥ 0.
∇ψ
b. Vary this integral expression for Eλ with respect to a small change ψλ → ψλ + δψ,
~ λ.
and determine δEλ as an integral expression quadratic in δψ and ∇ψ
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c. Using the completeness of the set ψλ , expand δψ =
µ cµ ψµ and integrate the
∗
expression for δEλ to obtain δEλ as a function of cµ , cµ , Eµ and Eλ .
d. Which ψλ (i.e., which Eλ ) can be determined by minimizing δEλ ?
2. Consider a particle of mass M constrained to move on a circle of radius a in the
x, y-plane.
a. Write down the Schrödinger equation in terms of the usual cylindrical-polar angle φ.
b. Determine the complete set of states, the corresponding energy spectrum and orthonormalize the stationary states.
c. Assume now that the particle has charge q and is placed in a small electric field
~ = Eêx . Determine the first non-zero perturbative correction to the energy levels.
E
~ = Bêz . Determine the first
d. Instead of the electric field, apply a small magnetic field B
non-zero perturbative correction to the energy levels.
e. What is the degeneracy of the unperturbed system (the one with E = 0 = B)? And
with E =
6 0 = B? And with E = 0 6= B?
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3. For a simple harmonic oscillator of mass m and spring constant k = mω 2 (where ω is
the classical frequency), the wave equation is
h̄2 d2 ψ 1 2
−
+ 2 kx ψ = Eψ .
2m dx2
a. Simplify the equation to −ψ 00 + y 2 ψ = 2²ψ, where primes denote differentiation with
respect to y: define x = ay and E = b² and the determine a, b.
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b. For this simplified equation, show that ψ approaches e− 2 y for large y.
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c. Let ψ = e− 2 y f (y), and determine the differential equation for f (y).
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d. Show that f must be either an even or an odd function of y.
e. Show that f cannot be an infinite power series of y.
f. Try f0 = 1, f1 = y and f2 = y 2 − c and determine c. Show that, corresponding to
these three solutions, ²0 = 12 , ²1 = 32 and ²2 = 52 ,
~ def
4. Consider the rôle of the angular momentum in quantum mechanics, where L
= ~r ×~
p
def
~ = −ih̄ ~r ×∇.
~
becomes L
~ generates infinitesimal rotations when applied to functions of ~r.
a. Show that L
b. Use this to show further that the angular momentum of a particle, in a central force
field, is conserved.
(Hint: rotation by a small angle δφ in the direction n̂ produces a ~r → ~r + δφ n̂ ×~r.)
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