Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Section 7.3.

There is a nice result derived in Shankar’s section 7.3 in which he shows that we can actually derive the ground state energy and wave function for the harmonic oscillator from the uncertainty principle. Classically, the energy of a harmonic oscillator is

where both and are continuous variables that can, in principle, take on any values. Thus classically it is possible for an oscillator to have giving a ground state with zero energy. In quantum mechanics, because and don’t commute, the position and momentum cannot both have precise values, which means that the ground state must have an energy greater than zero. This so-called *zero-point energy* is (as found by Solving Schrödinger’s equation)

To derive this without needing to solve Schrödinger’s equation, we first recall that a state in which the position-momentum uncertainty is a minimum must be a gaussian of form

where is a positive real constant, is the normalization constant, is the mean position and is the mean momentum. For a harmonic oscillator centred at , we have that both , so we know that the ground state wave function has the form

To normalize this we require (assuming is real)

Using the standard result for a gaussian integral (see Appendix 2 in Shankar or use Google)

Therefore

We need to find such that is minimized. The harmonic oscillator hamiltonian is

Since , the uncertainties become

Averaging 9 we get

At minimum uncertainty

so we have

The minimum energy can now be found by finding the value of that minimizes this function. Treating (not just ) as the independent variable, we have

This gives a minimum value for the mean energy of

To complete the derivation, we need to find the gaussian 4 that gives the correct value 19 for . That is, we need to find such that

This requires doing another gaussian integral:

We therefore get

which gives a normalized minimum energy wave function

This is the lowest possible value for the energy, but is it actually the ground state energy? What we have shown so far is that

where is the ground state energy. However, we can invoke the variational principle which states that if is any normalized function, then the ground state energy of any hamiltonian satisfies

Using we therefore have

Combining 29 and 31 we have

which means that

and therefore that , that is, 28 is actually the ground state wave function.

Although this clever little derivation gives us the ground state energy and wave function, it doesn’t say anything about the higher energy states, or tell us that they are all equally spaced with a spacing of . Nevertheless, it’s a pleasant exercise.