(Solved): 1D Harmonic Oscillator. A particle in a 1D harmonic oscillator potential with angular frequency = ...

1D Harmonic Oscillator. A particle in a 1D harmonic oscillator potential with angular frequency $?=2\mathrm{eV}/?$ is in a superposition of ground state and excited state wave functions. When multiple copies of the system are prepared and the energy of each system are measured, $60\%$ of the total number of copies gave an energy of $1\mathrm{eV},30\%$ of the total number gave an energy of $3\mathrm{eV}$, and the remaining number gave an energy of $5\mathrm{eV}$. Write down the wave function of the particle and calculate the uncertainty in the energy of the particle.