If frequency is defined as the cycles per time, then what is meant by "frequency of an electron"? If it refers to the rotation of electron around a nucleus, then which phenomenon is considered for a free electron i.e. an electron in a force field?
Is "frequency of an electron" an experimental quantity?
My teacher told me how to calculate the frequency of an electron. We started from finding energy of electron, then difference in energy, then we get this equation according to the Bohr radius of a hydrogen atom and
$$ f = \frac{z^2e^42\pi^2m}{h^3} \left(\frac{1}{n_1^2} – \frac{1}{n_2^2} \right)$$
Where:
- $z =$ atomic number
- $e =$ charge of proton
- $m =$ mass of electron
- $h =$ Planck constant
- $n =$ orbit number
From the last part of my equation, I am confused. Does the $n_1$ and $n_2$ show that that frequency will be the frequency of energy or electrons?
Best Answer
Since you used the tag wave-particle-duality, I imagine you mean the frequency $f$ that corresponds to an electron's energy $E$ via Planck's relation, $$E=hf,$$ where $h$ is Planck's constant. That is a valuable question and nothing to get picked on for. After all, if the electron is a wave with wavelength and so on, it surely has a frequency, right?
It turns out that this frequency is not very easy to measure. The reason for this is that the electron "wave" is usually complex-valued. That is, the thing that oscillates is a complex number $\psi=a+ib$, usually called its wavefunction. The real and imaginary parts of this wavefunction "rotate" into each other: $\psi$ will be real, then imaginary, then negative real, then negative imaginary, then real again, and so on and so on, in a continuous fashion. The frequency you're asking about is the frequency at which this happens.
Unfortunately, we are only ever able to directly measure the modulus of $\psi$, i.e. quantities of the form $|\psi|^2=a^2+b^2$, and this is constant even though $a$ and $b$ are oscillating. Schemes to try and measure $\psi$ in some (indirect) way are some of the most interesting measurements in quantum mechanics.
In this case there is a second problem which is also quite interesting, and it is the fact that only differences in energy can have physical meaning. Thus to ever measure the frequency$\leftrightarrow$energy of a particle, then we need to compare it with a second particle with a different frequency$\leftrightarrow$energy, and then measure the difference in frequencies$\leftrightarrow$energies. This will be present as a "beat" in the wavefunction, as we add together two complex numbers that are rotating at different frequencies, and it is in principle possible (though damned hard!) to measure.