Compare standing waves in a circle to those on a 1d string. What are the differences and similarities in the standing waves allowed?


Two waves are moving on a string. What must be true in order for them to form a standing wave?
  1. They must be moving in the same direction.
  2. They must have the same amplitude.
  3. They must have the same phase.
  4. They must have the same shape.
  5. They must be sines or cosines.
  6. They must have the same speeds.
  7. They must have the same period.



If ψ=1+2i, what is |ψ|2
  1. 1
  2. 5
  3. -4+4i
  4. 51/2
  5. 1+4i



What happens to the energies of electrons in a box as the box gets narrower?
  1. They get lower.
  2. They get higher.
  3. They stay the same.



The wave functions for particles in a rigid box between 0 and a are Asin(kx) with k = nπ/a. How would the wave functions change if the box were to go from -a/2 to a/2?
  1. The value of k would get smaller, but the wave function would have the same form.
  2. Some of the sines would change to cosines, but the k's would stay the same.
  3. Both k and the functional form would change.
  4. Nothing would change.



What would happen to the energies if an electron in a rigid box was replaced with a muon?
  1. The energies would go up.
  2. The energies would go down.
  3. The energies would stay the same.



A solution to the Schrödinger Equation for a rigid box with E = 0 is ψ(x) = Ax + B, where A and B are arbitrary constants. (This is a solution since ψ''(x) = 0.) Why would this energy not be acceptable?
  1. The wave function couldn't be expressed as a sum of sines or cosines.
  2. The wave function could not be normalized.
  3. It's impossible to have a zero energy.
  4. It's impossible to satisfy the boundary conditions.
  5. I really don't know.



Can a free particle have a well-defined momentum and energy?
  1. Yes. Any positive energy and any momentum are possible solutions to the Schrödinger Equation.
  2. No. A state with definite momentum or energy would not be normalizable.
  3. No. Only certain quantized values of momentum and energy are possible.
  4. It depends on the value of the potential energy.
  5. It depends on the value of the kinetic energy.



What is the probability density for finding a particle at the center of a rigid box of width a if it is in the third excited state?
  1. (2/3a)1/2
  2. 1
  3. (2/a)1/2
  4. 0
  5. (2/a)



How would you expect the energies of a particle in a finite square well potential to compare with the energies of a particle in an infinite square well potential for a given energy level?
  1. They would be the same.
  2. The energies for the infinite square well would be higher.
  3. The energies for the finite square well would be higher.
  4. It's impossible to tell.
  5. I don't know.



Which of the following functions are possible solutions to the Schrödinger Equation for a constant U(x)? (HINT: check by direct substitution)
  1. ψ(x) = eikx
  2. ψ(x) = cos(kx)
  3. ψ(x) = ekx
  4. ψ(x) = Ax2 + Bx



What are the states like in a finite square well when the energy exceeds the height U0 of the well?
  1. Such states do not exist.
  2. Those states can have any energy, since the particle is essentially free.
  3. Those states exist, but only for certain energies.
  4. I don't know.



An electron in a harmonic oscillator potential makes a transition between two energy levels, emitting a photon in the process. You can tell which states were involved in the transition from the wavelength of the photon.
  1. True
  2. False