Question

# Consider a particle in a one-dimensional **half **harmonic oscillator where V(x) = infinity for x

< 0 and V(x)= 1/2 mω^2x^ 2 for 0 ≤ x

(a) Plot the potential energy.

(b) Using your intuition, and the conditions that the potential energy puts on the wave function, sketch the three wave functions of lowest energy of this system.

(c) Plot the potential of the normal one-dimensional harmonic oscillator and sketch its six wave functions of lowest energy.

(d) Determine the eigenvalues and eigenvectors of the "half harmonic oscillator" Hamiltonian. Hint: How are they related to the eigenvalues and eigenvectors of the full harmonic oscillator?

(e) Compute the expectation value of the position operator for the ground state of the "half harmonic oscillator"