harmonic oscillator

asdf1

why is the lowest allowed energy not E=0 but some definite minimum E=E0?

siddharth

If you solve the Time Independent Schrodinger equation for the Harmonic Oscillator, that is
[tex] -\frac{\hbar^2}{2m} \frac{d^2\Psi}{dx^2} + \frac{1}{2}kx^2 \Psi = E \Psi [/tex]

The quantization of energy comes from the boundary conditions (ie, [itex] \Psi = 0 [/itex] when [itex] x= \infty [/itex] or [itex] x = -\infty [/itex]).

The permitted energy levels will be

[tex] E_n = (n+\frac{1}{2}) \hbar \omega [/tex]

So the lowest Energy is not E=0.

Galileo

I could give a hand-wave argument. We have E=1/2mv^2+1/2kx^2.
If E=0 both x and v are zero, which contradicts Heisenberg.

asdf1

thank you very much!!! :)