[camariela]

Hello,
Can someone please please help me out with this question:
I have to prove that the expectation value for the potential energy, V, for a harmonic oscillator in its ground state is 1/2 of the total energy E.

Thanks so much in advance,
camariela

[lemonoman]

I found a few different ways:

Calculation of Values with Equations and Stuff

This link http://theoserv.phy.umist.ac.uk/~brandes/QM/umist_qm/node88.html is a single-screen look at how much the potential and kinetic energies are for a harmonic oscillator....and low and behold, they are 1/2 the total energy of the system!  (1/2 × h(bar) × omega)

Seemingly ultra-rigorous Quantum Mechanics

If you take a look at sections VI. and III. of the following link:

http://classweb.gmu.edu/sdavis/chem332/ex11/ex11.htm#Expectation%20values%20of%20f(x)

Then take a look at the equation at the bottom of VI.  This shows what the eigenvalues for an energy function are.  In the case of this equation (Schrodingers) the energy is the eigenvalue(s).  And if you look through section III, this is the calculation of the expectation value.  Note that it is 1/2 of the energy value!


Hopefully these are what you're looking for!