What are the allowed energies for a quantum harmonic oscillator?

[Moham1287]

Hi all

I was just looking through my notes from my first year of my degree, and I couldn't find a missing bit. I know that Planck's postulate states that the allowed energies of a quantum simple harmonic oscillator are 0, hf, 2hf etc and that by the Schroedinger equation, you get E(n)=(n+1/2) hbar omega, but I can't explain why each of these give different answers. A quick google didn't bring up anything not password protected, and I don't have my textbooks with me at the moment. Anything to clear this up would be much appreciated!
Many thanks.

 

[jtbell]

I know that Planck's postulate states that the allowed energies of a quantum simple harmonic oscillator are 0, hf, 2hf etc

Planck was dealing with electromagnetic radiation, not the simple harmonic oscillator (mass on a spring and similar things).

 

[olgranpappy]

...the allowed energies of a quantum simple harmonic oscillator are 0, hf, 2hf etc and that by the Schroedinger equation, you get E(n)=(n+1/2) hbar omega, but I can't explain why each of these give different answers...

$$\hbar\omega=hf$$

...the remaining difference is just a constant ($$0.5\hbar\omega$$) which can be ignored.

 

[olgranpappy]

I.e., the Hamiltonians

$$H=\frac{p^2}{2m}+\frac{kx^2}{2}$$

and

$$H'=\frac{p^2}{2m}+\frac{kx^2}{2}-\hbar\sqrt{\frac{k}{4m}}$$

describe the same physics since they differ only by a constant... and furthermore, it should be obvious that they are equal in the classical limit.

 

[Gokul43201]

There's actually a little known paper by Einstein, where he first figures out that the Harmonic Oscillator needs a non-zero ground state energy, and he sticks in the 0.5 \hbar \omega by hand. Can't recall or pull up the citation. Anyone know what I'm talking about.

 

[premagg]

According to schrodinger equation,the particle cannot have zero energy.Because the solutions are (v+1/2)hf,v is vibrational quantum number.If v happens to be -1,then energy is -ve,considering zero potential the KE is -ve and so is momentum.This is not in correspondance with uncertainity principle.

 

[premagg]

jtbell is correct also,I am sorry to misunderstand your question.Planck was dealing with the blackbody radiations and schrodnger with quantum mechanical harmonic oscillator.