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ECE Solid State Device
Homework Assignment #2
2.1 The uncertainty principle is a cornerstone of quantum physics. The aim of this
problem is to verify it for a wave-function given by a Gaussian wavepacket (the ground
state of the parabolic potential). The wave-function can be written as

(x) = exp[-x2/42]/2
(a) Show that the probability corresponding to this wavefunction is normalized [5 pts]
(b) Find the averages <x>, <p>, <x2> and <p2> for this particle. We define the average of
an operator O as the symmetrized average
<O> = ∫[*OO**]dx/2,
the operator O acting on the quantities sitting to its immediate right. The operator for p is
–iħd/dx [5 x 4 = 20 pts]
(c) Find the standard deviations representing uncertainties in x and p
x =  (<x2> - <x>2),
p =  (<p2> - <p>2)
[5 x 2 = 10 pts]
(d) Show that the quantities x and p satisfy the minimum uncertainty principle
x .p = ħ/2 [5 pts]
Note that the Gaussian wavefunction is a minimum uncertainty solution. The Fourier
transform of a Gaussian is a Gaussian once again, meaning that the particle has the same
localized shape when evaluated in x space (position) or k space (momentum). This is
expected because the Hamiltonian for a parabolic potential is quadratic in both x and k,
and this symmetry gives us the minimum uncertainty of all potentials.
HINTS: You can use Mathematica/MathCAD to do the analytical integrals, or look up
Gaussian integrals online. But let me give some clues:
1. Integration of ODD functions of x between symmetric positive and negative
limits, e.g. plus or minus infinity, should be zero. A moment’s thinking will tell
you why that happens!
2. You should be able to write down the standard deviation of the Gaussian, by
inspection. After doing derivatives with x, you should be able to work out p also
with minimal effort.
2.2 The simple harmonic oscillator, ie, the parabolic potential from part 1, has equally
spaced eigenvalues that look like (n+1/2) ħ where  is the angular frequency of the
oscillator. Consider now the half-oscillator shown below, whose potential equals a
regular oscillator for x > 0 and equals infinity (hard wall) for x < 0. The hard wall
imposes additional boundary conditions on the regular oscillator solutions. From this
constraint alone, and the above information, draw the eigenvectors (‘modes’) of the halfoscillator and find its eigenvalues (you are welcome to run a numerical solution in
addition, just for confirmation). Explain the reasoning behind your analysis [10 points]
2.3. (a) Current density is ultimately determined by conservation of charge. If we define
charge density
n = *
then show using the Schrodinger equation that the current density operator
J = (ħ/2mi)(*d/dx – d*/dx)
satisfies the continuity equation, dn/dt = -dJ/dx, which represents charge conservation. (HINT),
when you do dn/dt, the time derivatives of  and * can be rewritten using the right side of
the Schrodinger equation) [4 pts]
(b) For a free electron described by a plane wave  = 0eikx, simplify the current
expression and interpret the result. Does this make sense? [3 pts]
(c) What is the current you get for a standing wave,  = 0sin(kx) ? [3 pts]