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
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
The total energy in state is so .