In the last video, we used the concept of a rate of spontaneous emission to estimate the average lifetime of excited states. We can, in addition, use this expression for the rate of spontaneous emission to come up with rules for the types of transitions that are allowed in atomic systems, specifically for the case of example hydrogen, we would in general need to evaluate matrix elements of this form to estimate the rate of transition between a state with quantum numbers nl and ml, and another state with numbers n prime l prime and ml prime. In particular, if this matrix element is equal to 0, then the rate of spontaneous emission is equal to 0, and this means that there's in general no transition between these two states. This is in the case of what's known as dipole transitions. There are other types of transitions allowed. You can have a magnetic, I guess this will be electric dipole transition, you can have magnetic dipole transitions and electric quadruple transitions, which are much, much weaker and therefore much less probable and can often be ignored. The idea then is to establish some rules to know in advance when this matrix element will be 0, and we won't go through the details, but in general, this matrix element is not equal to 0, when delta m, which ml, which we're defining by ml prime minus ml, so the change in the magnetic quantum number in the transition is either equal to 0 or equal to plus or minus 1. If this condition is not satisfied for the transition we're studying, then that transition is dipole forbidden. So there is no dipole transition between those two states. There's a second condition that says that delta ml, which we'll define as l prime minus l, has to be equal to plus or minus 1. These two conditions have to be satisfied for this transition to be allowed, and these are known as the selection rules. Notice that even in cases where these selection rules are satisfied, the matrix element of the dipole moment may still be 0 in some cases due to some other mechanism, but in general, there's always the possibility of these transitions satisfying these rules to be allowed. So if we apply this to the case of the hydrogen atom for the first four principle energy levels, we'll look at the core structure, we won't look at the fine structure for now. Yeah, l1, l2 and l3. Okay, so these are the possible energy levels for the core structure of hydrogen. And we can draw out the different the k-pads available for the excited states. This is the ground state, so it can't go anywhere. From this state, we can decay down to this state because l changes by minus 1. This state can't go anywhere. It can go the only choice it has for our lower energy level is over here, but the change in the orbital angular momentum doesn't satisfy our selection rules. If we look at this state now, this state is allowed to transition over here because the change in the orbital angular momentum satisfies our selection rule, and this is a lower energy state. This state over here can transition to this state because delta l changes by 1 and to this state. We're not focusing too much on the magnetic quantum numbers on the mls. These are part of, in the core structure, they're degenerate, but in general, they can only change by 0 or plus or minus 1. So within these, there's sub levels that we could also draw arrows to. If we look at this state now, this state can only decay down to this one. It can't go over here because that will require l to change by 2, which doesn't satisfy our selection rules. Likewise, this state can only decay down here. It's the only choice it has. The n equals 4 l equals 2 states. This one can decay down here because it changes l by 1, and it's a lower energy level and same thing for this one. This energy level, it can decay here where l changes by plus 1 or it can decay over here where l changes by minus 1. It can decay down here because then the l's wouldn't change. Finally, this state can only decay down to this one. It's a little crowded, but these are the allowed transitions for the core structure of hydrogen, according to our selection rules. This also illustrates the concept of multiple decay pathways. For example, if we were to estimate the lifetime of this state, we would need to calculate the rate of spontaneous emission from this state to this state and from this state down to this state. Add those up and calculate the lifetime based on that result. Now, also quickly note that while this state can't technically decay any further by a dipole transition, an electric dipole transition, there are other mechanisms that would let it decay in longer times because this is known as a meta-stable state, which can still decay, for example, by colliding with another atom or by multi-fold tundermation, which is a higher order decay process than the dipole transition.