Fall 2018 Homework Assignments
A particle of mass m is in the ground state of a one dimensional quantum well which extends from x=-a/2 to x=a/2 with wavefunction w(x). At time t=0, the size of the well is suddenly tripled (i.e. a → 3a), the well now extending from x=-3a/2 to x=3a/2. Due to the fast expansion of the well, the wavefunction at t=0 is still w(x), finite in the range from x=-a/2 to x=a/2: Ψ(x,t=0)=w(x). Find the energies and corresponding probabilities for measuring them after the expansion. What is the numerical probability that the a measurement of the energy of the particle after the expansion will be the same as that before the expansion.
1. A particle is in a superposition of two states of a harmonic oscillator potential: Ψ(t=0) = A ( uo(x) + u2(x) ) Where A is the normalization constant. Find the uncertainy of position x as a function of time: σx(t) = ? 2. A particle with mass m is moving in a one dimensional potential V(x) = α ( δ(x) + δ(x-a) ) where α is a constant. If the particle is incident from x = -∞ with energy E, find the transmission coefficient as a function of E and plot it for various values of α. (Hint: For obtaining computer solutions and plots, it is useful to work with scaled quantities which are unitless. For example, you could divide the Schrodinger equation by ħ2/(2ma2) - which has units of energy, to obtain unitless variables.)
1. Given that <x|f> = A (a-|x|)/a for |x| < a and zero otherwise <x|p> = exp(ipx/ħ)/√(2πħ) <p|g> = δ(p - po) find <f|g>. The quantities a, A, and po are constants. 2. Consider the 2x2 matrix A [ 1 -i ] A = [ ] [ i 1 ] and the vector V [ 1 ] V = [ ] . [ 0 ] (a) Find the eigenvalues and eigenvectors of A. (b) Expand V in terms of the eigenvectors of A. (c) Find exp(iA) V. (d) Find exp(iA). [Note that you can do part (c) without knowing exp(iA).]
The time independent Schrodinger Equation in three dimensions is (Px2 + Py2 + Pz2)/2m ψ(x,y,z) + V(x,y,z) ψ(x,y,z) = E ψ(x,y,z) where Pk = ħ/i ∂/∂xk. Consider now a cubic box whose one corner is at the origin, and three edges extend a distance a in the x, y, and z directions. The potential is zero inside the box and infinite outside. (a) Use separation of variables to show that ψ may be written as ψ(x,y,z) = X(x) Y(y) Z(z) and that the eigenstates of the energy operator are |nx ny nz > ↔ uE = (2/a)3/2 sin(nxπx/a) sin(nyπy/a) sin(nzπz/a) . (b) Use the state |Ψ> = ( |1 2 1> + i |2 1 1>)/√2 to find the expectation value of the z-component of the angular momentum operator Lz = xPy - yPx: <Lz>Ψ = <Ψ|Lz|Ψ>
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