# Can an atoms energy levels be changed by changing the de Broglie wavelength of electrons?

So an electron can only orbit a nucleus where its wavelength makes a standing wave, leading to discrete energy levels in atoms.

But the de Broglie wavelength of an electron = h/mv.

So the wavelength of an electron changes with velocity of an electron.

So are there different energy levels for electrons of different velocities (and wavelengths?) Or do all electrons in atoms have the same velocity? (and if so why?)

Thanks!

• ...quantum objects don't have velocities in the classical sense, and the deBroglie wavelength doesn't work for the atomic orbitals in the way you say. – ACuriousMind Aug 27 '15 at 1:27
• There are many questions about orbitals here on Physics SE. You should review them. Quantum mechanics is more interesting than simple Bohr orbital theory. – Jon Custer Aug 27 '15 at 1:52

## 3 Answers

So an electron can only orbit a nucleus where its wavelength makes a standing wave, leading to discrete energy levels in atoms.

That is the Bohr model which has been superseded by quantum mechanics. In quantum mechanics the electron occupies definite energy levels which arise because of the potential well that is generated by the nucleus. As the other answers say, it has not an orbit with a momentum, but an orbital, a locus in space that gives the probability density of finding it if trying to measure it.

But the de Broglie wavelength of an electron = h/mv.

This works for an electron as a free particle, not bound in a potential well where its probabilistic-wave nature dominates.

So the wavelength of an electron changes with velocity of an electron.

Of a free electron.

So are there different energy levels for electrons of different velocities (and wavelengths?) Or do all electrons in atoms have the same velocity? (and if so why?)

The electrons around the nucleus of an atom are in different energy levels that can be calculated by using the potential of the potential well. Due to the Pauli exclusion principle no two electrons can have the exact energy level .

To answer the title:

Can an atoms energy levels be changed by changing the de Broglie wavelength of electrons?

The energy levels are fixed by the potential and always have a width given by the wavefunction at that level.

• That's awesome - thanks so much for that clarification!!! – Riley Aug 27 '15 at 4:57

Well, you answered your question partly yourself. Only a finite amount of states (= energy and wavelength of the electron) are allowed, this is the basis of quantum mechanics. If an electron is in a certain state (ie. it has a certain energy and is in a certain orbital), you can then try to calculate its energy and wavefunction and from that its momentum (as ACuriousMind says, velocity is not the correct term to think about but you can calculate its momentum, or at least the value you expect the momentum to have, which is closely related to the classical notion of velocity).

When you then use quantum mechanics to calculate the expected value of the momentum, you in fact get that it's equal to zero because the electron can't go anywhere, it's bound to the nucleus! https://www.physicsforums.com/threads/expectation-value-of-momentum-in-h-atom.172343/

Please completely abandon the idea of an electron orbiting a nucleus. That is only believed in the now long defunct Bohr model of the hydrogen atom.

In modern Quantum Mechanics, in order to describe the electron in a hydrogenic (mono-electronic) atom, we need to solve the atom’s Schrödinger Equation which yields the wave functions $\Psi_{n,l,m}(r,\theta,\phi)$ of the electron.

These wave functions contains all the measurable information we can possibly know about the electron. But this information does not correspond well to Classical concepts like velocity or wavelengths. It’s tempting to consider the wave function $\Psi$ as a actual matter wave but in the most modern interpretations that is seen more as a (sometimes useful) analogy than an actual identity.

Measurable information about the electron, like energy, momentum or angular momentum, can be calculated by applying the relevant Quantum Operators to the wave function.

• Thanks! I'm only doing high school physics and trying to put together these two concepts we've been taught in class and these answers are a bit over my head... We're shown energy levels of (eg) hydrogen as 3.4 eV for n=2 (meaning two wavelengths fit) and so on. I don't quite get how that works or why it has to be discrete if the wavelength of an electron is determined by its 'velocity'...? maybe i'll just take it all at face value for now :s – Riley Aug 27 '15 at 3:47