Fall 2013 Homework Assignments
Find the complex Fourier series expansions of the functions below, which are periodic with length L. They are defined below for |x| < L/2 . The length a is less than L/2. 1) f(x) = ( 1 - |x| / a ) / a for |x| < a, zero otherwise. 2) f(x) = 3 ( 1 - x2 / a2 ) / (2a) for |x| < a, zero otherwise. Use the complex Fourier series coefficients cn that you have obtained above to plot the partial sums of these series, keeping terms with |n| < 10, 20 and 30. (Take L=1 and a=L/4 .) Repeat the plots for a = L/8. Comment on the differences you observe. Plot only the real part of the sum - the imaginary part should be zero anyway.
1) Determine the constants A for the following wavefunctions so that the corresponding probability density fuctions are normalized to unity: a) ψ1(x) = A exp( i k0 x) exp(- |x| / x0 ) b) ψ2(x) = A ( 1 - x2 / x02 ) for |x| < x0, zero otherwise where A and x0 are constants. 2) Find the Fourier transforms of the wavefunctions given in problem 1 above. 3) Find the expectation values <x>, <x2>, <p>, and <p2> for the wavefunctions in problem 1 and comment on the quantity Δx Δp .
1) Find the transmission coefficient for a δ-function potential of the form V(x) = α δ(x) , where α is a constant. 2) A particle of mass m is in an infinite potential well with a δ-function potential at the center so that { ∞ if x < -a/2 { V(x) = { α δ(x) if -a/2 < x < a/2 { { ∞ if a/2 < x Noting that the potential has the symmetry V(x) = V(-x), find the equations that will yield the energies of the bound states of the even and odd solutions.
1) Starting with the expression ∫ f* A g dx = ( ∫ g* A+ f dx )* , demonstrate that ( A B )+ = B+ A+ . 2) Find the expectation value < p2 > for any stationary state of the harmonic oscillator.
Consider the "displaced harmonic oscillator" Hamiltonian: H = p2/2m + k x2 + F x where is a constant (force). Rememeber that for F=0 this may be written as H = ℏωc ( a+ a- + 1/2 ) using the operators a± = ( ∓ ip/√[2m ℏωc] + x √[k/(2 ℏωc)] ) . Now, define the operators b± = a± + c (with c a constant) to show that H with F≠0 may be written as H = ℏωc ( a+ a- + 1/2 ) + ε0 . Find the values for c and ε0 .
Remember the vector representations of the spin eigenstates in the x, y, and z directions: <+z| ↔ (1 0) <-z| ↔ (0 1) <+y| ↔ (1 -i)/√[2] <-y| ↔ (1 i)/√[2] <+x| ↔ (1 1)/√[2] <-x| ↔ (1 -1)/√[2] 1) Find the probability of obtaining a result +ℏ/2 for the measurement of the z component of the spin, if the result of a previous measurement of the y component of the spin resulted in the value -ℏ/2. 2) Find the probability of obtaining a result -ℏ/2 for the measurement of the y component of the spin, if the result of a previous measurement of the x component of the spin resulted in the value +ℏ/2. 3) Express the spin state |ψ> = a |+y> + b |-y> in terms of a) spin eigenstates in the x direction, b) spin eigenstates in the z direction.
1) Consider the matrix 0 1 0 1 0 1 0 1 0 Find its eigenvalues and and normalized eigenvectors. 2) Expand the vector 1 1 x 1/√[3] in terms of these eigenvectors. 1
Consider the matrix 0 1 0 A = 1 0 1 0 1 0 Find sin( A ) .
Wavefunction of a particle is given as Ψ(x,y,z) = [ α(x2 + y2) + βz2 ] exp(-x2/x02) . Express this wave function as an expansion in terms of the angular momentum eigenfunctions, i.e. as a sum of terms of the form R(r) Ylm(θ,φ), where Ylm(θ,φ) are the spherical harmonics.
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.