2003.11. 7
1.Blackbody Radiation
Using the Planck distribution law, 8πh(ν/c)^3
ρ(ν) = ───────
e^(hν/kT) - 1
find the wavelength of maximum emission as a function of temperature which
is known as the Wien displacement law. (15%)
(h=6.6260755×10^-34 Js, k=1.380658×10^-23 JK^-1)
2.de Broglie's Matter Wave
What is the de Broglie wavelength of a nitrogen molecule at room temperature
? Compare this to the average distance between nitrogen molecules in a gas
at 1 bar at room temperature. (15%)
3.Uncertainty Principle
Estimate the zero-point energies for the following systems by using the
Heisenberg uncertainty principle ΔxΔp ≧ h/2:
a)A particle confined in a box of length a.
b)A particle in a three dimensional box with sides of length a,b and c.
c)A harmonic oscillator with frequency ν. The total energy for a simple
harmonic oscillator is given by E = (p^2)/2m + k(x^2)/2 , where
k = 4 π^2 m ν^2 .
(Hint:You need to treat the position uncertainty(say ξ~Δx) as a
variational parameter. Minimize E(ξ) with respect to ξ to obtain the
best result. The final result of zero-point energy should have both
kinetic and potential contributions. )
d)A particle constrained in a ring with radius R. Instead of using
uncertainty principle, solve this problem directly from Schrodinger
-
h^2 d^2
equation: －─── ─── Ψ(ψ) = EΨ(ψ)
2m R^2 dψ^2
with boundary condition, Ψ(ψ+2π) = Ψ(ψ)
Express your answers in terms of the mass of the particle m, Planck
-
constant h and relevant length scales for each particular system. You
have to show your reasoning for each problem to get full credits.
(20%)
4.Harmonic Oscillator
The 2-dim harmonic oscillator has the potential-energy function
1 2 1 2
V = ─ k x + ─ k y , where the k's are two force constants. What is the
2 x 2 y
expression for the quantym mechanical energy levels? Find the degree of
degeneracy of each of the four lowest energy levels in the isotropic case,
i.e. kx=ky=k (15%)
5.Variation Principle ^
If we use Ψ as a trial wavefunction for Hamiltonian H, where
ψ0 + aψ1
Ψ= ─────── and ψ0 and ψ1 are the two lowest eigenfunctions with
(1+a^2)^(1/2)
engenvalues E0 and E1 (with E1>E0), show that the variational method yields
a=0 and corresponding optimal eigenvalue is E0. (10%)
6.Hydrogen Atom
What is the magnitude of the angular momentum for the electrons in 3s, 3p
and 3d orbitals? How many radial and angular nodes are there for each of
thest orbitals. (10%)
7.Helium Atom
a)State the Pauli exclusion principle in terms of the interchange of
identical particles. (5%)
b)Construct the approximate total wavefunction that is consistent with Pauli
exclusion principle for the ground state of helium using orbital
approximation. What is the atomic term symbol for the ground state of
helium atom. (5%)
c)Find the wavefunctions and the corresponding term symbols for the excited
states of helium atom that can be constructed by the product of 1s and 2s
hydrogenlike orbitals. Which are the lowest excited states? Briefly
discuss the effect of external magnetic field on the energies of these
states. (15%)
Total:110%
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