Download Probability Distributions and Wavefunctions in Quantum Mechanics - Prof. Glenn Ciolek and more Assignments Physics in PDF only on Docsity!
239B - Probability Distributions
- Consider the situation of a particle for which the probability density for finding a particle in the region 0 < x < L is given by. The probability of finding the particle outside this region is zero. a) Plot or sketch this distribution between x = 0 and x = L. b) What is the probability of finding the particle between x = 0 and x = L? (You need to explicitly do the integral or make a solid mathematical argument. Show your work.)
P =
L
∫ 0 L
:. P of finding the particle between 0 and L is 1. c) What is the probability of finding the particle between x = 0 and x = L/2? (Hint: You might be able to do this without doing the integral. Think about symmetry.) Using symmetry P of finding the particle between 0 and 0.5L is 0.5. d) What is the probability of finding the particle between and where = 10-20 L. Your calculator will probably fail you here because we chose the region to be extremely narrow. (Hint: You might be able to do this without doing the integral in a formal way. Think about the idea of the Riemann sum in Calculus.)
(sin^ (^
πL
L ))
2
40A - Wavefunctions
- Several possible forms for the spatial form of the wavefunction for a particle are sketched below. List the ones that are not physical, and give a reason for why each one is not. Dashed lines indicate that the wavefunction jumps from one value to the next at a single point. 2 – There are undefined slopes at jumps, so it is not physical 3 – There is undefined slope at the end jump, so it is not physical 4 – The graph fails the vertical line test, therefore it is not a function, so it is not physical 6 – The graph has undefined slope at the midpoint, so it is not physical
2) Consider the wavefunction Ψ ( x , t )= ψ ( x )( cos ωt + i sin ωt ) where ψ ( x ) is a real function. Calculate
the probability density for this wavefunction in terms of the variables given in the equation.
3) a) Show that ψ ( x ) = A sin kx is a solution to the time-independent Schrodinger Equation with potential
equal to zero:
2
2 m
2
∂ x
2 = Eψ^ and find the relation between^ k^ ∧ E^ that solves the equation.
A B C
D E
F
4) An electron is in the n=10^9 energy level of a strangely shaped quantum well so that the
wavefunction ψ^ ( x^ )^ has the approximate
form shown below. (The value of the
wavefunction is zero outside of the limits
of the graph.) Note that the function at
x = L / 2 is steep, but it is not
discontinuous (the same is true for it’s
derivative as well).
a) Sketch the probability density that is consistent with the wavefunction shown. Include a scale
in terms of A.
b) Assuming that the probability of finding the particle in the space between 0 and L is unity,
find the value of A.
c) What is the probability that the electron will be found in the left-hand side of the box? (Your
answer must be consistent with your sketch in part b. It may either be in terms of A and
L, or it may be a number.) Explain your logic.
A
0 L
40B – Heisenberg Uncertainty Principle - Again
The Heisenberg Uncertainty Principle states that it is not possible to simultaneously measure the position and momentum of a particle with absolute certainty. The mathematical statement of the principle in the textbook is:
∆ x ∆ p = σx σ p ≥ h / 4 π
One of the approaches to understanding the uncertainty principle is to think about how to add waves of different wavelengths to one another to create a peaked waveform.
1) Consider a particle for which the spatial part of a wavefunction is ψ ( x ) = A e − a^ x
2 where A and a are real, positive constants. a) If the value of a is increased, what effect does this have on the uncertainty in the position of the particle? Explain. b) If the value of a is increased, what effect would this have on the uncertainty in momentum of the particle? Explain.
2) Electrons of kinetic energy K 0 are shot through a very narrow slit of width L and are detected when
they hit a phosphor screen a few meters away. A diffraction pattern is observed in the probability
distribution of arriving electrons with a central maximum of width ywidth. If the kinetic energy of the
electrons is reduced to one-half its initial value, how does the width of the diffraction pattern change?
