(a) A black, spherical deep space exploration craft has a radius of 1m, and continuously dissipates 10W of power. What is the surface temperature of the craft? What could be done to the surface of this craft so that with the same power dissipation, the surface temperature remains at 300K?
(b) X-rays may be produced by accelerating electrons through a potential difference, and then hitting a target. (The decelerating electron radiates.) If the potential difference used is 10KV, at what range do you expect the wavelength of the X-rays to be? Are there any minimum or maximum values?
Consider the complex numbers u=x+iy, v=z+it and w=z exp(it) where x, y, z, and t are real variables.
(a) Express the following quantities in the form A+iB where A and B are real:
u*v
u/v
u¾
w2
uv
wv
exp(u2)
sin(u)
tanh(u)
ln(u)
(b) Now, express the same quantities above in the form R exp(i θ) where R and θ are real.
Simplify your results as much as you can.
(a) Consider the periodic function f(x)=f(x+λ), defined in one period as
{ 0 if -λ/2 < x < 0
f(x) = {
{ 1 if 0 < x < λ/2
Find the coefficients cn of the complex Fourier series expansion of this function.
Plot the functions you obtain from this series, when you truncate the series such
that |n| ≤ 1, 3, 5, and 11. What is happening near x=0?
(b) Find Fourier transforms of the following functions:
f(x)=A exp(-|x|/L)
f(x)=A cos(kx)
f(x)=A exp(-|x|/L) cos(kx)
f(x)=A exp[-(x/L)2] sin(kx)
For each of the transforms, estimate roughly the uncertainty in x and in k values
and discuss the value of the product of these two uncertainties.
(c) Which of the functions in part (b) can be probability distribution functions?
Normalize these functions and find the expectation values <x> and <x2> corresponding to them.
Normalize the wavefunctions given below and determine the quantitities
<x>, Δx, <p>, Δp, and ΔxΔp for each case:
(a) Ψ(x)= A exp(-|x|/L + i k0 x)
(b) Ψ(x)= A if x0 < x < x1, zero otherwise
(c) Ψ(x)= A exp( i k0 x) if x0 < x < x1, zero otherwise
(d) Φ(k)= A exp[ -x02 (k - k0)2 + i k x1 ]
(e) Φ(k)= A k exp( i k x0 ) for 0 < k < k0, zero otherwise
Notes: (1) A, L, x0, x1, and k0 are constants.
(2) For evaluating derivatives of discontinuous functions above, smoothen
the discontinuities as shown below, where a is a small parameter.
Do problem 8 at the end of chapter 4 of the textbook (the problem of a particle
initially in the ground state of a box of size a/2, which then suddenly expands
to size a). In addition, plot the quantity |Ψ(x,t)|2 as a function of x for
several values of t:
To avoid putting in numerical values for physical quantities, use scaled, dimensionless
variables. For example, plot the graphs as a function of x/a, and display the results
at times t/τ = 0, t/τ = 1, t/τ = 2, t/τ = 3, and t/τ = 4, where τ is a fraction of the
oscillation period corresponding to the energy difference between the ground state
and the first excited state:
h / τ = ( E2 - E1 ) / 4
Finally, calculate the value of τ for an electron in a quantum well of width a = 10nm.
Given a particle in a one-dimensional quantum box with a δ-function singularity at the center:
{ α δ(x-a/2) for 0 < x < a
V(x) = {
{ ∞ otherwise,
where α is positive, corresponding to a repulsive potential.
Find the wavefunctions and the energies of the ground and first excited states.
(You will need to express the wavenumber as the solution of a transcendal equation.)
Hint: In order to get a symmetric potential, it may help to shift the origin so that it coinsides
with the δ-function.
Consider the angular monetum operators Lx, Ly, Lz, and L2 = Lx2 + Ly2 + Lz2. Work out the following commutators: [Lx,L2] , [Ly,L2] , [Lz,L2] [Lx,V(r)] , [Ly,V(r)] , [Lz,V(r)] [L2,V(r)] where V(r) represents a potential which is a function of the radial coordinate with r2 = x2 + y2 + z2. Express the physical implications for commutators which are zero. Which components of the angular momentum are conserved if the potential energy has the form V(R,z) where R2 = x2 + y2 ?
Consider two electron states, with wavefunctions Ψ(x)= A exp [ -(x-a)2/x02 ] and Ψ(x)= A exp [ -(x-b)2/x02 ] where a, b, and x0 are constant distances. (a) Normalize the wavefunctions. (b) Construct the symmetric and antisymmetric space wavefunctions for two particles in these states. (c) Find the expectation value < (x1 - x2)2 > for the two types of wavefunctions. (d) Discuss the above result for the cases when the particles are close together ( |a-b| ≈ x0 ) and when they are far apart ( |a-b| >> x0 ) .