In this video, we will introduce Heisenberg picture. We've seen unitary operators that are used to change basis set. In this case, we're not really changing the state ket. We're not doing anything to the state ket, but we're just changing the way in which we represent the state ket, changing the basis set. But there are also unitary operators that actually change the state ket, such as translation, rotation, and time evolution. In these cases, we are actually doing something to the quantum state. As a result, the quantum state changes. If it helps you understand these two situations a little better, the change of basis set is like rotating the coordinate system. Consider a position vector in a three-dimensional space. So you're representing the position vector in terms of X, Y, Z coordinate. What you're doing is you're choosing a set of basis vectors, X unit vector, Y unit vector, and Z unit vector and expressing your position vector in terms of those X, Y, Z unit vectors. Now, suppose that you rotated the coordinate system, you rotated the X, Y, Z unit vectors. Now, it will change the coordinates of the position vector but the position vector itself remains the same. You're not really doing anything to the position vector. That will be the case of this first bullet but in the second bullet is like you are actually rotating the position vector itself. In that case, obviously, the vector changed and therefore the coordinates of the vector changed even as you're not really doing anything to your coordinate system, your basis vector. That will be the case for this second bullet. Let's denote a time evolution operator U and the time evolution of vector cap vector Alpha is represented by just acting these time evolution operator onto this original vector. This is the time evolution. You are changing the quantum state. This time evolution operator we've seen is unitary and we can easily see that unitary operator, unitary transformation preserves inner product. The inner product between Beta and Alpha remain the same even after you apply unitary operation U and U dagger to these ket, Alpha, and bra Beta. Because by definition, unitary operator is an operator where the Hermitian adjoint dagger is equal to its inverse. Now, let's consider a time evolution of a matrix element of an operator X. Matrix element, if you recall by definition, is the X operator multiplied with a bra Beta from the left, and then ket Alpha from the right. That just is a definition. Let's apply a time evolution operations to this. Then you evolve these bra Beta over time, and that is represented by this U dagger operation and then you also evolve this Alpha ket over time, and that's represented by this operator U here. Now, because of associativity, you can express this general matrix representation evolve over time as this. In this case, X operator remains the same but your quantum state, represented by this bra Beta and ket Alpha evolves over time. Or you can consider the operator X is changed by this similarity transformation while Beta and Alpha quantum state remain the same. Obviously, these two cases are equivalent and in one picture here, operators don't change over time. Only the states bras and kets change over time. In the other picture, only the operator change over time and that time evolution is given by the similarity transformation while the quantum states represented by these kets and bras remain unchanged. The first interpretation is called Schrodinger's picture. First interpretation meaning only the state kets change over time and operators don't. That's the Schrodinger picture. This is what we've been using so far. But the other interpretation where the state kets don't change over time while the operators change over time is called the Heisenberg picture. In classical physics, if you recall, we don't use state kets to describe the physical system. We only use observables, which are represented by operators in quantum mechanics. In classical physics, we deal with time evolution of observables like momentum in an energy. In that sense, Heisenberg picture could provide a closer connection with the classical physics, because in Heisenberg picture we deal with time evolution of operators which represent physical observables. Now, we can express the state kets and observables in Schrodinger picture and Heisenberg picture. Recall the time evolution operator U, is given by this exponential function of H. In Heisenberg picture, the time evolution of the operator A is given by the similarity transformation. The operator at t equals zero initial state, is simply the operator in Schrodinger picture initial state, which doesn't change over time. This similarity transformation defines the relationship between the operator in Schrodinger picture time-independent and the same operator in Heisenberg picture time dependent. The obvious implication is that at t equals zero, the Heisenberg picture and Schrodinger picture basically gives you the same operator. Similarly for state ket, in Heisenberg picture the ket doesn't change over time, so that should be equal to the initial state of that same ket in Schrodinger picture. The time evolved ket in Schrodinger picture, simply should be given by applying time evolution operator to the initial state ket, which is the state ket in Heisenberg picture. Now the expectation value should remain the same. Expectation value shouldn't depend on which picture you use. We can calculate the expectation value of operator A in Schrodinger picture, which is time independent and the state ket Alpha here in Schrodinger picture is time-dependent. Now they use this equation here and this equation that relates the kets and bras in Schrodinger or Heisenberg picture, also the operators in Schrodinger or in Heisenberg pictures. Write it out this, then use the fact that the operator U is unitary, so U there go times U, U there go times U, simply gives you an identity operator. That gives you the expectation value in Heisenberg picture, and they are the same as they should. Now consider the time derivative of the operator A in Heisenberg picture. Remember in Heisenberg picture, operator is time-dependent, so you can take a time derivative and see how it evolves over time. By definition, the Heisenberg picture operator is related to the Schrodinger picture operator through this similarity transformation. If you take a derivative, noting that the Schrodinger picture operator is time independent, so there is no time derivative for that, you only need to take a derivative of U dagger and U, as shown here, use the chain rule. Recall that this operator U is an exponential function of the Hamiltonian operator H. If you take a derivative of U dagger, you get i times H Hamiltonian divided by h bar. Then if you could take a derivative of U, then you get negative i over h bar times the operator H. Now, if you recall that these U dagger A of SU, this is the operator A in Heisenberg picture, same thing here. Then you can write it like this. It's just the commutator bracket between the A operator in Heisenberg picture with the Hamiltonian operator H. You can just change the order and put a negative sign. You can write it like this. This equation we just derived specifies the time evolution of an operator A in Heisenberg picture, and the time evolution of an operator is given by the Hamiltonian operator as it should. Just as the time evolution in Schrodinger picture is also specified by the Hamiltonian operator. In either case, Schrodinger picture versus Heisenberg picture, time evolution is always related to the Hamiltonian operator. The time evolution of an operator in Heisenberg picture is given by this Heisenberg's equation of motion. Whereas the time evolution of state kets in Schrodinger picture is given by the Schrodinger's time-dependent equation. Now, this Heisenberg equation looks very much similar to the Poisson bracket in classical physics. This is somewhat expected because, in classical physics, we don't use statins. We only deal with time evolution observables, just as in Heisenberg pictures, we initially expected some close similarity between Heisenberg picture and classical picture, and this Heisenberg equation of motion and the Poisson bracket equation in classical physics gives you one example of that. Now, let's consider a time evolution of base kets. State kets are stationary in Heisenberg picture, operators are time-dependent. However, base kets, the kets that we're using as a basis set, they are time-dependent. Why? Because these guys are eigenkets of a Hermitian operator, the base kets are. In Heisenberg picture, this Hermitian operator is time-dependent. Naturally, the eigenket should be time-dependent. A state ket on the arbitrary ket describing a quantum state in Heisenberg picture can be represented as a superposition linear combination of these basis kets that is time-independent. Base kets themselves are time-dependent when you make a superposition of that time-dependent base kets to express a state ket that result is time-independent in Heisenberg picture. Based kets are time-independent in Schrodinger picture, why? Because the operators are time-independent and therefore, their eigenkets should be time-independent. Let's consider an eigenket a prime on a Hermitian operator A, eigenvalue is a prime. In Schrodinger's picture, A is stationary time-independent, and therefore, these a prime, a prime eigenvalue are all time-independent. In the Heisenberg picture, however, you're A operator is time-dependent, which is related to these initial state or the Schrodinger picture operator through this similarity transformation. What we can do is we apply, we take the eigenvalue equation of A shown here. You multiply U dagger on both sides of the equation from the left that you get this. Then you insert U times U dagger which is Identity Operator in-between A operator and the eigenket a prime. Now, what we can do is using associativity, you notice this U dagger AU is the a operator in the Heisenberg picture. Then you're left with this ket, U dagger a prime ket. Then on the right-hand side, same eigenvalue, a prime, and U dagger a prime ket. The eigenvalue equation is preserved. The state ket satisfies the same eigenvalue equation is just that in the Heisenberg case, both the operator here and the eigenkets are time-dependent as they should. Now, expansion coefficients should be the same in both pictures. In Schrodinger's picture, the ket Alpha as a function of time is given by this equation here. These coefficients here is time-dependent because base kets remember are time-independent. The quotients are simply given by the inner product of the base ket and the ket in question, Alpha. Because Alpha evolves over time through these time evolution operator U, you can express Alpha of t as this. In short, in Heisenberg picture, your Alpha is expressed by the same equation here. It is Alpha in Heisenberg picture is time-independent and the coefficient here is given by the same way, you just take an inner product between Alpha and the one of the base ket. That gives you the same equation. The Alpha prime t is expressed in terms of the time evolution, is expressed in terms of U. If you compare these two equations, you can see that they're identical.
