The result of applying the translation operator e−iaˆp/ℏ on the wavefunction f(x) is:

A one-dimensional quantum harmonic oscillator with frequency ω is in its ground state. Its normalized wave function is given by ψ(x) = (mωπħ)1/4 exp[-mω2ħ x2]. If the frequency is suddenly increased to 2ω, the probability of finding the particle in its new ground state is:

A quantum mechanical particle in a harmonic potential has the wave function 1√2 [ψ0(x) + ψ1(x)] at t = 0, where ψ0(x) and ψ1(x) are the ground state and first excited state wave functions respectively. If the oscillator frequency is ω, the probability density of finding the particle at x after time t = πω is:

If L̂ is the angular momentum operator for a quantum particle, then L̂ × L̂ is equal to:

For a particle in the angular momentum state |l = 4, ml = 2⟩, the expectation value of the operator LxLy is:

A spin-12 particle is placed in a magnetic field B = Bxx̂ + Byŷ with spin Hamiltonian Ĥ = -AŜ · B (where A > 0 and Ŝ is the spin operator). The eigenvalues of the Hamiltonian are: