- #1

tom.stoer

Science Advisor

- 5,766

- 161

## Main Question or Discussion Point

The photoelectric effect is usually presented as an example disproving classical electromagnetism as viable model for interaction of light with matter and as evidence of quantization of energy in the electromagnetic field, i.e. the existence of photons. I would like to discuss a thought based on non-relativistic quantum mechanics w/o and relation to Planck, Einstein etc. showing - imho - why this is not so straightforward.

Using a simple 1-dim. model with a single electron plus time-dependent perturbation theory - in the same manner as used in the derivation of the spectrum of the hydrogen atom including selection rules - we obtain the following picture:

We have

- a quantum well of finite depth -V

- a discrete spectrum for [itex]\epsilon_i < 0[/itex]

- a continuous spectrum for [itex]\epsilon_f > 0[/itex]

- a

Calculating the transition matrix element

[tex]M_{fi} = \langle \epsilon_f|E(x,t)| \epsilon_i\rangle \sim E_0\,\delta(\epsilon_{fi} - \hbar\omega)[/tex]

and using Fermi's golden rule we find that

- the probability is proportional to [itex]|E_0|^2[/itex]

- therefore the number of photoelectrons is proportional to that probability

- the energy of the photoelectron is [itex]\epsilon_f = \hbar\omega - |\epsilon_i|[/itex]

- therefore the initial energy plays the role of the so-called "work-function", i.e. [itex]W = |\epsilon_i|[/itex]

That means that time-dependent perturbation theory of non-relativistic quantum mechanics with a

Replies welcome ...

Using a simple 1-dim. model with a single electron plus time-dependent perturbation theory - in the same manner as used in the derivation of the spectrum of the hydrogen atom including selection rules - we obtain the following picture:

We have

- a quantum well of finite depth -V

- a discrete spectrum for [itex]\epsilon_i < 0[/itex]

- a continuous spectrum for [itex]\epsilon_f > 0[/itex]

- a

*classical*electromagnetic field [itex]E(x,t) = E_0\,\sin(kx - \omega t)[/itex]Calculating the transition matrix element

[tex]M_{fi} = \langle \epsilon_f|E(x,t)| \epsilon_i\rangle \sim E_0\,\delta(\epsilon_{fi} - \hbar\omega)[/tex]

and using Fermi's golden rule we find that

- the probability is proportional to [itex]|E_0|^2[/itex]

- therefore the number of photoelectrons is proportional to that probability

- the energy of the photoelectron is [itex]\epsilon_f = \hbar\omega - |\epsilon_i|[/itex]

- therefore the initial energy plays the role of the so-called "work-function", i.e. [itex]W = |\epsilon_i|[/itex]

That means that time-dependent perturbation theory of non-relativistic quantum mechanics with a

*classical*electromagnetic field is able to reproduce the essential characteristics of the photoelectric effect w/o ever mentioning light quanta. Therefore this effects shows clearly that the interaction of light with matter cannot be obtained from purely classical reasoning, but it also shows that Einstein's hypotheses of light quanta cannot be derived in a straightforward manner.Replies welcome ...