Let me preface by saying that "coupling" is a favorite physicist word that is perhaps best described linguistically than rigorously; it's deployed in a few different situations.

In general, we say that a coupling exists in quantum mechanics if the evolution of one part of the system depends on another quantity, which could be either classical or quantum. I'll give one example for each.

Suppose the Hamiltonian of a two-level system is an internal Hamiltonian $H_\mathrm{int}$ and an additional part that depends on some external parameter, maybe $\theta$:
\begin{equation}
H = H_\mathrm{int} + \theta \sigma_z
\end{equation}
Here the $\sigma_z$ was arbitrary. The point is that this system's evolution depends directly on the parameter $\theta$--maybe it's an external magnetic field, or some other feature of the environment. In this case, we would generally say that the system is "coupled to $\theta$." (You'd often see this in a metrological context, where we might be interested in using a quantum system coupled to an external parameter to measure the parameter.) In this case, there is only one quantum object, evolving under $H$.

Another common system--maybe a little more general--would be the evolution of two different variables both treated quantum mechanically. The idea here is that there would be some operator $A$ characterizing one observable of interest, and another $B$ characterizing a second. Then the Hamiltonian might be:
\begin{equation}
H = H_A + H_B + H_{AB}
\end{equation}
Where $H_A$ doesn't contain any term depending on $B$, $H_B$ doesn't contain any term depending on $A$, and $H_{AB}$ might have terms like $A \cdot B$, $A^2 B$, etc. The reason why this couples the system is that if we now evaluate Heisenberg equations of motion $\dot{A} = \frac{i}{\hbar} \left[ H, A \right]$ we'll find that the $H_{AB}$ term will put terms depending on $B$ into $\dot{A}$ and vice versa. Therefore, solving the equations of motion will require describing both $A$ and $B$. On the other hand, if the equations "decouple" or we do something to decouple them ourselves, we can usually find a solution for $A(t)$ that doesn't depend on $B$ and vice versa.

This is all paralleled in classical mechanics, by the way, where we would call two variables coupled if they appeared in each others' equations of motion.

EDIT: Peter Shor points out that objects can be "indirectly coupled," which is correct but would usually require me to introduce another variable $C$. I think the most general statement of being coupled/uncoupled is asking whether the equations of motion can be solved independently of each other.

We have for pressure $P$ the thermodynamic identity

$$
P=-\frac{dU}{dV}\bigg|_S
$$

Where $U$ is the interal energy, $V$ is volume and $S$ is entropy. Of course, these are ensemble quantities which are not even defined for a single particle. We can perform a formal calculation: replacing $U=E_n$, the nth eigen-energy, and ignoring $S$ we have

$$
P_n=-\frac{d}{dV}E_n
$$

In the nth energy eigenstate, *with $l=0$*, $E_n=\frac{n^2\pi^2}{2mR^2}$, $\hbar=1$. Note that $R=R(V)$ since $V=\frac{4}{3}\pi R^3$. Using the chain rule

$$
P_n=\frac{n^2\pi^2}{m}R^{-3}\frac{dR}{dV}=\frac{n^2 \pi}{4mR^5}
$$

It should be stressed that this calculation is by formal analogy rather than from a microscopic thermodynamic theory. It is also *not* a deduction purely from single particle quantum mechanics.

If we let $P=F/A$, with force $F$ and area $A$, the above does agree with what your prof said. Again, note that force is not even defined in quantum mechanics.

## Best Answer

Sounds like graphene physics or something similar. You won't find it in a dictionary.

In the band structures of many materials, it is common to find multiple similar points in reciprocal (momentum) space. For example, in silicon's band structure there are six distinct conduction bands that all have similar behaviour. These six points came to be known as valleys. Among other details, the "valley degeneracy" of 6 is an important factor to take into account when calculating electronic properties of silicon.

Graphene's band structure has two distinct bands. These are often known as the K and K' valleys, and they are centered around the K and K' points in reciprocal space. They are very symmetric and also are closely related to a spin-like property of the electrons in the graphene, known as pseudospin. In essence, you can imagine the K valley as being "pseudospin up" and the K' valley as being "pseudospin down". An electron can also be in a superposition of pseudospin up and down, so you can imagine "pseudospin left", "pseudospin right", and so on. The resemblance with spin comes from the fact that scattering processes between the two valleys are fairly rare, and so pseudospin is conserved over some distance. Also, it means that besides the normal factor of 2 degeneracy from real spin, the electrons in graphene have a further factor of 2 degeneracy from the valleys / spin. On the other hand, pseudospin is not associated with a magnetic moment, unlike real spin.