[isears]

I'm working through MIT OCW's online chemistry course and I had a question about one of the exercises from the textbook.

In this exercise, we calculate the Energy of an electron ejected from the incident EM radiation (photoelectric effect) by using the electron's velocity and the formula for kinetic energy (0.5mv^2). In addition, we consider the electron as a wave and calculate the electron's wavelength by using the velocity (again) and the formula for wavelength given momentum (λ = h / mv).

I found that the electron had a non-trivial wavelength (on the scale of nanometers) and took a step further by calculating the electron's energy by using the formula E = hc / λ.

The result was two different values for energy (1) KE derived from 0.5mv^2 and (2) energy derived from hc / λ. Why is this? Why do we calculate one value for the energy of an electron when we consider it as a particle and another value when we consider it as a wave?

Furthermore, when I attempt to use the value for the electron's energy to calculate the metal's work function, given the value for the incident EM radiation, only the calculated KE gives the correct answer. If I try to plug in the value for energy calculated from wave formulas it doesn't come out right. So in that context it seems like KE is the "right" value and the value calculated from the wave formula is somehow "wrong".

Would appreciate any insight into these apparently conflicting energy values!

[Borek]

For which particles does the E = hc / λ formula hold?

[Enthalpy]

Yes, take the relation for an electron.

Sidenote: a piece of metal has many electrons at very different energy levels. By using the work function, you implicitly supposed that the extracted electron came from the Fermi level, and the assignment probably made the same mistake.

But this isn't generally the case. If the photon has enough energy to leave the freed electron with speed, then it can perfectly extract an electron from below the Fermi level, and then the remaining kinetic energy is smaller. It's the most common case.

There is more. In every interaction, energy is conserved, and momentum too. The free electron has an energy-momentum relation, the electron in a metal has a completely different one and may have a strong momentum if its energy fits. The photon brings little momentum, so a phonon has to to it, but can be rare.

[isears]

Does E = hc / λ only apply to photons?

I guess that would make sense, given the use of the constant c (speed of light).

Can E = hc / λ still apply for other particles if the constant c is substituted for the particle's actual velocity? I plugged it into my equations and the numbers didn't work out, so I'm guessing E = hc / λ strictly applies to particles moving at the speed of light.

@Enthalpy: Yeah, I think the exercise made the assumption that the electron whose velocity was observed was the easiest electron to extract. You raise an interesting point that I had not considered.

Thanks much for the responses!

[Borek]

Does E = hc / λ only apply to photons?

Yes.

[Enthalpy]

[...] the electron had a non-trivial wavelength (on the scale of nanometres) [...]

Nanometres are reasonable for an electron and eV-range energy. The size of an atom, in the order of 100pm, is an electron wavelength, though said wave is an evanescent wave over much of the volume - call it a tunnel effect if you prefer so.

More massive particles have shorter wavelengths. For instance interference fringes are harder to obtain with them, but this was made with fullerenes two decades ago, with proteins more recently.