Consider an electron in an infinite square well. The expectation values of momentum and angular momentum are all zero for energy eigenstates. An electron transition is accompanied by the emission or absorption of photons. And we know the momentum of a photon is hk and the angluar momentum (spin) is 1.
The momentum and angular momentum should be conserved in the transition process. Does that mean all electron transitions between energy levels are prohibited in such system?
If such transitions are prohibited, how do we explain light-emitting quantum dots?
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Thanks for the answers and comments below. Now I realize a quantum dot is more like an atom than a hollow box. But if we consider the theoretical ideal square well, there seems to be no simultaneous eigenstates of both energy and momentum – at least I can't figure it out.
Consider a 2D quare well. The energey eigenstates can be shown as:
Energy eigenstates in a 2D Box
Apparently they are not momentum eigenstates. Since these states form a complete set, we can set a momentum eigenstate to be:
$|k_x\rangle = \sum_{n_x,n_y} C_{n_x,n_y} |n_x,n_y\rangle$
where C's are constants, and
$P_x |k_x\rangle = \hbar k_x |k_x\rangle$
Keep going on:
$P_x |k_x\rangle = -i\hbar\nabla_x \sum_{n_x,n_y} C_{n_x,n_y} |n_x,n_y\rangle = -i\hbar\nabla_x \left( C_{1,1} \sin(k_{1}x)\sin(k_{1}y)+\cdots\right)$
$= -i\hbar\left( C_{1,1} k_1 \cos(k_1 x)\sin(k_1 y)+\cdots\right)$
$=???\; \hbar k_x \left( C_{1,1} \sin(k_{1}x)\sin(k_{1}y)+\cdots\right) = \hbar k_x |k_x\rangle$
I don't know any way to transform cos*sin's to sin*sin's without messing them up.
Is there anything wrong with my calculation?
Best Answer
In an ideal 3-D square well the electron's movement along the three axes is independent. This is not true for a real quantum dot which has a spherical shape (close to it) and is thus more similar to hydrogen atom. As soon as there is a coupling between movement along $x$,$y$ and $z$ the electron will start orbiting around the quantum dot and will posess an angular momentum.
Now, a linear conjugated molecule can be considered as a 1-D square well. Linear carbon chains have been discovered in the interstellar space and also produced in the lab. To interact with a photon such molecule has to have a transition dipole moment which means that the electron cloud should move. If we don't consider $p$-orbitals (that would add a second dimension), only a non-symmetric molecule like $C_6N$ can have a nonzero transition moment and such molecules are in fact very strong light absorbers.