Problem 1:

Electrons of kinetic energy 10 eV travel a distance of 2 km.  If the size of the initial wave packet is 10^-9 m, estimate the size at the end of their travel.

Problem 2:

The wavefunction y of a particle is written as a linear combination of the three orthonormal eigenfunctions {f_i} of the observable A with eigenvalues a_i (i = 1,2,3).

Find <A>.  What is the probability that the measurement of A yields a₂?  Find the wavefunction immediately after this measurement.

Problem 3:

In relativistic mechanics, energy and momentum of a free particle are related by the expression E² = p²c² + m²c⁴.  Construct the relativistic analogue of the Schroedinger equation by introducing the appropriate operators.

Problem 4:

Consider a three-state quantum mechanical system with an orthonormal ‘color’ basis {|R>, |G>, |B>} (‘red,’ ‘blue,’ and ‘green’ respectively).  Its evolution is governed by the Hamiltonian

 .

(a)  Construct the matrix representation of this Hamiltonian using the {|R>, |G>, |B>} basis.
(b)  Find the energy eigenvalues and normalized eigenstates of the system.  Express the latter as linear combinations of |R>, |G>, |B>.
(c)  At time t = 0 the state vector is |y(0)> = |G>.  Find the state vector |y(t)> at an arbitrary time t.
(d)  After starting from the initial conditions of (c), the ‘color’ is measured at time t = t₀ and found to be green.  What are the probabilities for the color to be measured as red, green, or blue at time t = 2t₀?

Problem 5:

Assume the wave function of a particle is
  
Here a and p₀ are real constants and N is a normalization constant.
(a) Find N so that y(x) is normalized.
(b) If the position of the particle is measured, what is the probability of finding the particle between and ?
(c) Calculate the mean value of the momentum of the particle.

Problem 6:

Consider the operators whose action is defined by the equations below:

Find the commutator [O₁, O₂].

Problem 7:

(a)  What is the wavelength of a 10 eV electron and what is the energy of a photon with this same wavelength?
(b)  Light with a wavelength of 300 nm strikes a metal whose work function is 2.2 eV.  What is the shortest de Broglie wavelength for the electrons that are produced as photoelectrons?
(c)  A surface is irradiated with monochromatic light whose wavelength can be varied.  Above a wavelength of 500 nm, no photoelectrons are emitted from the surface.  With an unknown wavelength, a stopping potential of 3V is necessary to eliminate the photoelectric current.  What is the unknown wavelength?