Does an electron have kinetic energy when attached to a proton?

[Lolicon]

does an electron have kinetic energy when attached to a proton? if not, what is it transformed into?

 

[PeroK]

does an electron have kinetic energy when attached to a proton? if not, what is it transformed into?

A proton and electron bound together is called a hydrogen atom. The hydrogen atom has a set of possible energies, each of which is the sum of the kinetic energy and potential energy.

 

[Dale]

Yes, it has KE

 

[etotheipi]

There is a nice result called the Virial theorem which says that if the potential energy of interaction between two bound particles goes as $V(r) = \lambda r^n$, then the potential and kinetic energies of that system are related via $2 \langle T \rangle = n \langle V \rangle$. For an electrostatic interaction that potential has $n=-1$, so $2\langle T \rangle = - \langle V \rangle$. Equivalently, the total energy is $E = \langle T \rangle + \langle V \rangle = - \langle T \rangle = \langle V \rangle/2$.

The energy eigenvalues of a hydrogen atom look like$$E = \frac{-13.6 \text{eV}}{n^2}$$For instance at the ground state, $n=1$, then $E = -13.6 \text{eV}$, $\langle V \rangle = -27.2 \text{eV}$ and $\langle T \rangle = 13.6 \text{eV}$. As you go up energy levels, the potential energy and total energy increase, whilst the kinetic energy decreases.

 

[PeroK]

The energy eigenvalues of a hydrogen atom look like$$E = \frac{-13.6 \text{eV}}{n^2}$$For instance at the ground state, $n=1$, then $E = -13.6 \text{eV}$, $V = -27.2 \text{eV}$ and $T = 13.6 \text{eV}$. As you go up energy levels, the potential energy and total energy increase, whilst the kinetic energy decreases.

Those are, of course, the expected values of $T$ and $V$ for a QM system.

 

[etotheipi]

Those are, of course, the expected values of $T$ and $V$ for a QM system.

Thanks, give me two seconds and I'll put the langles and the rangles where they belong

 

[jtbell]

Possible nitpick about terminology: "expectation value" is the term that I've always seen in English-language QM textbooks, journal articles, etc., at least in the US.

To me, "expected value" has the connotation "It has a definite value, and this is what I expect it to be." Of course, that's not what we're talking about here.

 

[PeroK]

https://en.wikipedia.org/wiki/Expected_value

Expected value, expectation, mean, average. Whatever!

 

[hilbert2]

Whenever an electron is moving in some limited space, as when orbiting an atom or put in a "box", it has a nonzero expectation value of kinetic energy.