Harmonic Oscillator

The given state is a linear superposition of the second and fourth normalized harmonic oscillator wave functions, i.e., and , which are energy eigenstates of the quantum harmonic oscillator. The energy eigenvalues corresponding to these states are:
- Energy of state is ,
- Energy of state is .
For the state , the time-dependent wave functions are: At , we have: Now, for the wave functions to be orthogonal at some time , the phase difference between the two states must be an odd multiple of . The phase difference between the two energy eigenstates is: For orthogonality, we require: Thus, the wave function will be orthogonal to at time , which corresponds to option (A).