Fall 2010 Homework Assignments
1) Determine the matrix elements corresponding to the spin operators Sx, Sy, and Sz in the basis which makes Sy diagonal. 2) An electron is known to be in the spin state |α> = (1/3) |+z> + c |-x> where the kets |+z> and |-x> are eigenkets of the Sz and Sx operators respectively. o Find the constant c, assuming that it is real and positive. o Find the expectation values of Sx, Sy, and Sz operators for this state.
1) Given that <x|α> = ψ(x) and <x|β> = φ(x), show that quantities <α|X2|β> and <α|P2|β> may be expressed as integrals over ψ and φ. 2) Given that the functions ψ and φ in the above problem are ψ(x) = A exp(-γ|x|) and φ(x) = B / (x2 + δ2) with A, B, γ, and δ constants, o determine the normalization constants A and B assuming that they are real and positive o determine the quantities <p|α> , <p|β> and <alpha|β> (If you cannot evaluate any of the integrals, you can look them up from a table.)
Given that at time t=0, <x|α,0> = A exp ( - (x/x0)2 + i k0 x ) where A , x0 and k0 are constants. 1) Find the normalization constant A. 2) Find the fluctuation in the X measurement ΔX. 3) Find <p|α,0>. 4) Find the expectation value of momentum <P> and its fluctuation ΔP. What is ΔXΔP ? 5) Assume that this state corresponds to a free particle. Then, remember that momentum eigenstates are also energy eigenstates. What is <p|α,t> for t > 0 ? 6) What is <x|α,t> for t > 0 ? 7) Find the expectation value of X for this state for t > 0 .
1) Consider a particle in a one dimensional harmonic oscillator potential. Find the time-dependent expectation values of the following operators, if it is known that at time t=0 the particle is in the state ( |0> + |1>) / (2)1/2. |0> and |1> represent the lowest and first excited eigenstates of the particle. X2 P2 XP2 + P2X X2P + PX2 cos(kX) where k is a constant 2) Evaluate the following commutators containing X, P, and the harmonic oscillator operators. Terms in your result may contain the identity, number, and only one of a or a+ operators. In your result, carry the a operators to the right hand side and the a+ operators to the left so that the terms look like, for example, a+NN or Naa. [X2,P] [N,a+a+] [N,a2] [aNa,NaPaN] [XaNaX,PaX]
1) Let H = P2/2m + V(X) be the total energy operator for a one dimensional quantum system with discrete eigenstates H |a> = Ea |a>. Show the following results: (a) Σa'|<a|X|a'>|2 (Ea' - Ea) = ℏ2/2m . (b) <a|P|a'> = im/ℏ (Ea - Ea') <a|X|a'> and hence Σa'|<a|X|a'>|2 (Ea' - Ea)2 = ℏ2/m2 <a|P2|a> . 2) A particle of mass m is in a one dimensional potential V(x) = vδ(x-a) + vδ(x+a) with v<0. (a) Find the wave function for a bound state with even parity, i.e. Ψ(-x) = Ψ(x). (b) Find an expression for the energy for even parity states and determine how many such states exist. (c) Find the scattering solution when the particle is incident with energy E > 0 on the potential. Determine the reflection and transmission probabilities (if necessary, numerically) as a function of E, v and a.
1) Find the density matrix and the ensemble average of the Sz operator for the following mixtures of particles: (a) Pure state of particles in the |x+> state. (b) Pure state of particles in the |x-> state. (c) Mixture of particles in the states |x+> (with ratio 9/25) and |x-> (with ratio 16/25). (d) Pure state of particles in the ( 3 |x+> + 4 |x-> )/5 state. (e) Equal mixtures of particles in states ( 3 |x+> + 4 |x-> )/5 , ( 3 |y+> + 4 |y-> )/5 , and ( 3 |z+> + 4 |z-> )/5 . 2) Thermal electrons are being emitted in the x direction by a source with energy probability density P(E) = A exp(-E/kBT), where A is a normalization constant. (a) Assuming that the electrons are free particles once they are emitted, what is the density matrix (in the coordinate basis)? (Take into consideration that electrons are being emitted only in the +x direction.) (b) Find the ensemble expectation value [P] of the momentum operator for these electrons. (For this part it will be easier to use the density matrix [and the momentum operator!] in the energy basis.)
Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.