Quantum Harmonic Oscillator: Ground State Solution
we try the following form for the wavefunction
Substituting this function into the Schrodinger equation by evaluating the second derivative gives
For this to be a solution to the Schrodinger equation for all values of x, the coefficients of each power of x must be equal. That gives us a method for fitting the boundary conditions in the differential equation. Setting the coefficients of the square of x equal to each other:
Then setting the constant terms equal gives the energy
This is a very significant physical result because it tells us that the energy of a system described by a harmonic oscillator potential cannot have zero energy. Physical systems such as atoms in a solid lattice or in polyatomic molecules in a gas cannot have zero energy even at absolute zero temperature. The energy of the ground vibrational state is often referred to a "zero point vibration". The zero point energy is sufficient to prevent liquid helium-4 from freezing at atmospheric pressure, no matter how low the temperature.
Schrodinger equation concepts