**Required math: calculus**

**Required physics: 3-d Schrödinger equation**

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 4.40.

Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Chapter 13, Exercise 13.1.5.

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We’ve seen the virial theorem in one dimension, which states:

where is the kinetic energy.

We can derive the 3-d version of the virial theorem using a similar method. From the formula for the rate of change of an observable, we have,

assuming that the potential is time-independent. (This is what Shankar refers to as Ehrenfest’s theorem.) In three dimensions, we have

Since each term in the commutator (except for the potential ) contains only one of the three spatial coordinates, any derivative term commutes with any other derivative term that contains a different variable. The remaining three non-zero commutators, one for each coordinate, can be calculated in the same way as in one dimension. We are therefore left with a simple generalization of the result for one dimension.

For stationary states the time derivative is zero, so

For hydrogen,

so since ,

Thus we have

But we know that the total energy for the hydrogen atom in quantum state is so we get and .

For the 3-d harmonic oscillator

so

The total energy in state is so .

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