In this video, we will use the time-dependent perturbation theory to calculate transition probability. Let's first consider a simple example of a constant perturbation potential turned on at a certain time, which we set equal to 0. At t equals 0, this constant time-independent potential V is turned on. Before that, the potential was 0. This is our V of t. The 0 for the equation simply gives the initial condition. In this case, we assume that the quantum system was initially in state i, so C_n^0 is simply this chronicle Delta. If n is equal to i, then it's 1, and if n is not equal to i, then it's 0. That defines our initial condition. The first-order correction or the first-order coefficient C_n^1 is given by this formula that we derived using the Dyson series, and in this case, this V_ni is time-independent. Because this V here is a time-independent constant, we can take that outside of the integral, and we simply need to integral this exponential function, which gives us this quantity here. Now, the transition probability then is found by taking the absolute value square of the coefficients, and the algebra is straightforward, and we get this equation here. In the denominator out in front of the sine square function, we have this energy difference squared, and inside the sine squared function, the frequency of oscillation is also related to the energy difference between the two states. I here is the initial state and n is the state that the electron or the quantum system is transitioning into. If you look at this function, then it has a peak at Omega equals 0 or Omega_ni equals 0, which means the E_n, the final state energy is equal to the initial state energy. That point has the highest probability, and as you deviate away from that condition, the probability goes down, the maximum amplitude here scales as t squared, and the width scales as 1 over t. Now, as t becomes very large, we wait long enough so that the quantum system makes a complete transition or finishes the transition into a new quantum state. The transition probability is appreciable only when the t is approximately equal to this quantity here. From this, we can derive this equation here by defining Delta t being the duration during which the perturbation is turned on. The Delta t, the time duration during which the perturbation is on, and Delta E, which is the energy difference between the two quantum state, that product is roughly h-bar. This equation looks very much like the uncertainty relationship that we found for other variables, for example, position and momentum uncertainty, and this equation is actually called oftentimes time energy uncertainty. However, we should notice that this uncertainty relation is fundamentally different from the position momentum uncertainty, because time here is not a measurable variable, and not at least in non-relativistic quantum mechanics. Time here is just a parameter. This equation, rather than uncertainty or the inability of precisely determined certain observable variables, this uncertain relation rather sets the timescale of the transition. When the time t, Delta t during the perturbation is on, is very short than the transition can afford to have a fairly large uncertainty or deviation away from the energy conservation condition whereas the Delta t, the time duration of the perturbation is long, then the energy conservation requirement is more strictly applied. That's the physical meaning of this equation. When we have E_i is equal to E_n, then the coefficient is simply given by this, and it is proportional to this t squared quadratic dependence on time. That may seem unreasonable and we will have to do something to correct this dependence. Let us consider an example of helium atom in the excited state. Initial state, both of these two electrons in the helium atom is in 2s orbital. After the transition, one of the electron falls back down to the ground state, 1s, but the other electron is released from the atom and become a free electron. This type of transition is called the Auger transition, and the free electron released as a result of this transition is called the Auger electron. In this case, what we're interested in is the total probability given by the sum over all possible final state with energy E_n, which is roughly equal to E_i. What we want is not just one particular c_n1 absolute value squared, but all of those coefficients summed over all possible state with this energy constraints. But notice that one of the electron is in free space, and there, we don't have a discrete set of energy, rather we have a continuum of allowed energy. We need to convert this summation into an integral. In order to convert this summation into an integral, we need to define a quantity called the density of state. The density of states Rho of EdE is defined as the number of allowed states, number of quantum states within an infinitesimal energy interval between E and E plus dE. With that definition, we can convert the summation here into an integral. We integrate c_n1 absolute value square, this guy here, over energy weighted by this density of state, because at a given energy, there may be more than one quantum states available. That is accounted for by this density of state. Now we plug in the expression that we obtained for c_n1 in the previous slide, and we obtain this equation here. Now we notice that in the limit of t is going to infinity, this quantity inside the integral becomes this Delta function. The amplitude of this Delta function is linearly proportional to t. Now with this, we can evaluate this integral which is this. Now the sine square in this denominator, e_n minus c_i absolute Delta squared, that becomes the Delta function shown here and we take this guy here, the matrix element of the perturbation potential absolute value squared. We take that outside the integral by taking the average. We can do that because of the nature of the Delta function we are only interested in a relatively narrow range of energy. We can assume that this has a very weak energy dependence and treat it as a constant. Take it outside by taking the average value. Then the integration is very straightforward because of this Delta function. We simply need to evaluate the density of state at the energy equal to the initial energy. This time we obtain a transition probability that is linearly proportional to t. Now this allows us to define then a quantity called transition rate, which is the transition probability per unit time. You can think of it as the number of transitions the quantum system makes per unit time. The transition rate is simply the time derivative of the transition probability. The transition probability is linear in time, so it simply gives you this and this equation is the famous for me is golden rule which gives you the rate of transition for a quantum system induced by this perturbation Hamiltonian. Sometimes this transition rate is written like this without the density of state, but even in this case it should be remembered that this expression is actually derived by integrating over a range of energy. Now, finally we can proceed to the second order correction and the equation that we derive for the second order correction using the time dependent perturbation theory as shown here. Once again, these V_ mn and V_ mi are all time independent constant because our perturbation potential is time independent. We can take them outside the integral. The integration becomes this and we combine this second-order correction with the first-order correction, we obtain the transition rate. This here is the first-order term and this here is the second-order term. If you look at the second-order term, it includes two transitions. It is made of two transitions. First i, initial state to m state, and then from m to n. The transition from initial state i to final state n in the first-order it goes directly. We simply evaluate the matrix element of the perturbation potential between initial and final state and that allows us to determine the transition rate. In the second order correction, what we do is we consider an indirect transition going through this intermediate state m and we sum over all possible intermediate state to calculate the total transition rate. This non-energy conserving transition from i to m, some intermediate state and m to n, the final state are called the virtual transition and this is a characteristic of a second-order transition in quantum mechanics.
