Practice Problems for Exam 2#
Compute the average momentum of harmonic oscillator.
Compute average position of electron confined in 1D box of size L.
How would degeneracy of first excited state of 3D cubic box change when you increase z dimension two fold.
Calculate energy of harmonic oscillator in ground state given spring constant \(k=12 N/m\) and mass \(0.5 kg\)
Determine the probability density function of a particle in a box of length \(L\) in the first excited state \(n=2\) at a point \(x = \frac{L}{4}\).
Determine the wavelength of the photon needed to excite an electron from the ground state to the first excited state in a 1D box of length \(L = 1 nm\)
Calcualte average kinetic energy of particle in a box given that it is found 90% of the time in ground and 10% time in the first excited.
Write down time-dependent wavefunctio for a particle in a box of the previous problem.
Write down explicit integral for computing the probablity of electron to be found in left third part of the box.
Which operators are not Hermitian \(x\), \(x^2\), \(-i\frac{d}{dx}\), \(-i \frac{d^2}{dx^2}\), \(d/dx\)
You made a measuerment of two components of momentum and found their values. What can we say about Operators corresponding to these observables?
What is \(\langle 1|\psi\rangle\) equal for a quantum state described by \(|\psi\rangle\) which is characterized by superpsoition of frist four states of harmonic oscillator with equal probabilities.
Which ground state of electron will be lower if it is placed in box \(L_x=2L_y=3L_z\) vs \(L_x=3L_y=2L_z\)
Calculate Energy of a 3D symmetric harmonic oscillator in ground state. Take \(\hbar\omega=1\) for all dimensions.
Given operator \(d/dx\) and its two eigenfunctions \(|0>\) and \(|1>\) will \(\langle 0|1\rangle\) be zero?
In units of m=1, h=1, L=1 the 3D particle in a box has energy \(E=112\) determine degeneracy of the energy level and list all quantum numbers.