In this video, we will discuss Hamiltonian momentum and position operators and the uncertainty relationships between these operators. First we define Hamiltonian operator as shown here. We have seen this in the discussion of time-independent Schrodinger equation. So adopting this definition for Hamiltonian time independent Schrodinger equation can be written in a compact form as an eigenvalue equation for the Hamiltonian operator. In the non-relativistic quantum mechanics, Hamiltonian operator is related to the total energy of the quantum system. The Hamiltonian operator is also related to the time evolution of the quantum state. This equation here is called time dependent Schrodinger equation, and we will discuss this in a lot more detail later. For now, we simply introduce this simple equation as time-dependent Schrodinger equation, and show that the Hamiltonian operator also dictates the time evolution of the quantum states through this equation. The momentum operator is postulated to be this differential operator as shown here. If we define momentum operator like this, the classical definition of kinetic energy, which is p squared divided by 2m, gives you the 1st term of the Hamiltonian. Therefore together with the 2nd term of the Hamiltonian, which is the potential energy, the Hamiltonian properly represents the total energy of the system. Now we notice that the plane wave exponential i times k vector times r position vector. That is an eigenfunction of the momentum operator. Because if we apply momentum operator here, then these differential operator simply pulls out i times k and gives you these h bar k, a number scalar times the same exponential function. So the eigenvalue here, h bar times k is the eigenvalue. This is consistent with the De Broglie's matter wave hypothesis where the p momentum is inversely related to Lambda. If we adopt the standard definition of k wave number, which is equal to two Pi over Lambda. Notice here, the p here, is a three-dimensional vector, not an operator. This is basically a number in some sense as opposed to the operator p and this quantity here acts as an eigenvalue. Position operator is trivial in the framework of using wave functions of positions to describe quantum state, position operator is simply the position vector itself. To simplify we omit the hat symbol and simply just write r as the position operator. For the z component of the position, the operator is just the variable z. Now, we can discuss the uncertainty relation between the position and momentum operator. Specifically, we're going to consider the commutation of x component of momentum and position x. Now, if we consider the commutator as shown here, and apply that to an arbitrary ket Alpha. Then the commutator is, by definition, P_x times x minus x times P_x. If you use the definition of operator P as the differential operator, you can write it out like this. Then for this term here, we apply the chain rule and spell it out like this. This term, and this term cancels out, but this term survives, gives you this. This equation that we derived in the previous slide applies to any arbitrary ket alpha, we can write it as this. The commutator of p_x and x is equal to negative ih bar and you compare that with the general commutator relationship that we discussed in the previous video. In this case, C operator here simply is equal to negative h-bar. In that case, we can reduce this general uncertainty relationship that we derived in the previous lecture as this. Now, this involves delta p square and delta x square and h-bar square, so if you get rid of the squares, then the uncertainty of p momentum and uncertainty of position product is always greater than h-bar over two. If you use instead of momentum k, then the uncertainty in k vector or k wave number and the uncertainty in position product is always greater than 1/2. Now, you can do the same thing with the energy and time, and as we mentioned earlier, we use the Hamiltonian operator here as the time evolution operator. From the time-dependent Schrodinger equation, h operator here can be represented as the time derivative multiplied with a ih bar. If we calculate the commutator between h and t time, then you get a very similar equation as the momentum and position case and you still get i times h-bar as a result, which gives you the same uncertainty relationship. So the uncertainty of energy and uncertainty of time, product is always greater than h-bar over two. If you use the expression of energy as h-bar times omega angular frequency, then it leads to the well-known frequency time uncertainty relationship; delta omega times delta t product is greater than 1/2. Now, the frequency time uncertainty relationship is well-known in Fourier analysis. You cannot have both well-defined frequency and well-defined time for a signal. When the signal is a very short pulse, it necessarily contains a wide range of frequencies and vice versa. Minimum frequency bandwidth for a given time interval is always determined by the same uncertainty relationship that we just derived for the energy and time uncertainty relationship. Position and momentum uncertainty relationship is also well-known and can be derived the same way from the Fourier analysis, it is a very well-known phenomenon in wave theory as well. For example, if you do a diffraction experiments through a slit, then the wave passing through a narrow slit diverges, the smaller the slit width is, the more divergent the beam emerging from it. The slit width determines the uncertainty of position, narrow slit means small uncertainty in position and that leads to a very large uncertainty in momentum along that direction, and that leads to a diverging beam. The smaller the slit is, the more divergent the emerging beam is, and this is a direct consequence of the momentum position uncertainty relation. If you recall, the very first lecture in this course dealt with electron beam diffraction experiments, the double slit experiments using electron beam. In order to properly interpret the experimental results, understanding the momentum position uncertainty relationship is crucial.
