Download 453 Introduction to Quantum Mechanics (Winter 2005)

453 Introduction to Quantum Mechanics (Winter 2005)
Practice exam for the final
1. Consider a particle in the infinite square well potential


 0,
if 0 ≤ x ≤ a
 ∞,
otherwise
V (x) = 
(0.1)
Show that the possible values of the energy are given by
En =
n2 π 2 h̄2
.
2ma2
(0.2)
Instruction Show your work, make sure you include the Schrödinger equation, its
solutions satisfying the natural boundary conditions and how they lead to quantization.
2. In deriving the Generalized Uncertainty Principle we consider two observables
A and B and their corresponding uncertainties given by
σA2 = < (Â− < A >)Ψ|(Â− < A >)Ψ >=< f |f >
(0.3)
σA2 = < (B̂− < B >)Ψ|(B̂− < B >)Ψ >=< g|g >
(0.4)
where
|f >≡ (Â− < A >)|Ψ >,
and |g >≡ (B̂− < B >)|Ψ > .
(0.5)
The necessary and sufficient condition for achieving minimum of uncertainty (minimum wave packet) is
|g >= ia|f >,
a real.
(0.6)
Show that for position-momentum uncertainty the criterion (0.6) implies that the
minimum-uncertainty wave packet is a Gaussian, i. e., the wave function that minimizes the uncertainty principle is a Gaussian.
3. Show that if f is simultaneously an eigenfunction of L2 and of Lz , that is,
L2 f = λf,
and Lz f = µf,
then the square of the eigenvalue of Lz cannot exceed the eigenvalue of L2 .
(0.7)
4. An electron is in the spin state


 3i 
.
χ = A
(0.8)
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i) Determine the normalization constant A.
ii) Find the expectation value of Sx and Sz .
5. Suppose you had three particles in a one-dimensional harmonic oscillator potential, in thermal equilibrium, with total energy E = (9/2)h̄ω. If they are distinguishable particles (but all with the same mass),( i) what are the possible occupationnumber configurations? (ii) What is the most probable configuration?
6. In the two-dimensional infinite potential well the energies are given by
Enx ny
π 2 h̄2
=
2m
n2x n2y
+ 2 .
lx2
ly
!
(0.9)
Assuming that the number of free electrons per unit area is σ, calculate the Fermi
energy for electrons in a two-dimensional infinite square well.
7. The most prominent feature of the hydrogen spectrum in the visible region is
the red Balmer line, coming from the transition n = 3 to n = 2. i) Determine the
wavelength and the frequency of this line.
The fine structure, that is, the inclusion of the spin-orbit coupling and the consideration of the relativistic correction yield a correction to the Bohr energies of the
form
4n
En2
3−
.
=
2
2mc
j + 1/2
!
Ef s
(0.10)
ii) Determine how many sublevels the n = 3 level splits into and find Ef s for each of
them.
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