In this video, we'll discuss time-dependent Schrodinger equation. We just introduced time-dependent Schrodinger equation that's shown here without derivation. In the left-hand side, we have Hamiltonian acting on a ket or a wave function, just the same as the time-independent equation. On the right-hand side, instead of the energy eigenvalue, we have these time derivative on the right-hand side. The Hamiltonian, of course, is defined the usual way. These differential operator representing the kinetic energy and the potential energy. Here, the potential energy, of course, can, in general, depend on time, however, in this course, we will restrict our attention to time-independent potential which covers a large number of important problems. We noticed that this is not an eigenvalue equation, so unlike the time-independent equation where we are to solve an eigenvalue equation meaning that we have to determine both the eigenfunction and eigenvalue, this is not an eigenfunction and this equation simply allows us to predict the state at certain time t given the initial condition at t_0. We can solve these first-order differential equations in time simply by this exponential function here if we can define exponential function for an operator. You can't convince yourself that if we define the exponential operator for this Hamiltonian operator the usual way, as done in algebra in an infinite series expansion as shown here, then this actually is a solution of the time-independent Schrodinger equation and this is a solution to the time-independent equation. This operator here, this exponential operator containing H, is called the time-evolution operator because it acts on a state at t equals t_0 and it evolves a state to a time t in the future. Now, the time evolution operator is unitary, so if you take Hermitian adjoint to this exponential operator, well, what you'll do is you'll take a complex conjugate, so this negative i becomes positive i, and then you will also take Hermitian adjoint for this Hamiltonian operator, but Hamiltonian operator is our mission, so it is equal to itself and therefore this operator simply is the inverse of the original operators. Hermitian adjoint is equal to its inverse. This is a unitary operator. Now, let us consider the energy eigenstate. Energy eigenstate by definition satisfies this time-independent Schrodinger equation or the eigenvalue equation for Hamiltonian. If you substitute this time-independent equation onto the time-dependent equation, the left-hand side, it simply becomes the energy eigenvalue times the energy eigenket. Right-hand side remains the same, and this is a very simple first-order differential equation and you can immediately solve it to as shown here. The solution to this time-dependent Schrodinger equation for energy eigenvalue E_n is simply given as this. The time evolution of the energy eigenket is the initial state times this exponential phase factor. Now, energy eigenket if you remember of a Hamiltonian operator, forms a complete set. What that means is that for any arbitrary state ket alpha, you can express it as a linear combination of the energy eigenkets and this set of quotient C_n uniquely defines this an arbitrary state Alpha. If you evolve this state Alpha, arbitrary ket, then all you have to do is to attach this exponential phase factor to each of the energy eigenket. Because according to the Schrodinger's time-dependent equation, each eigenket evolves with this phase factor, so just attach the phase factor to each energy eigenket. Then that collection, that series of c_n coefficient times this phase factor, that collection of these combined time-dependent coefficients now determines the state Alpha at time t. More in general, if you want to write this Alpha at time t as a linear combination of this energy eigenket, remember this Alpha evolved in time is also a ket, a vector in the Hilbert space, and that vector should be also expressible as a linear combination of the energy eigenket which forms a complete set. The coefficients now is different of course, because this Alpha of t is not the same of this original Alpha at t equals 0, so the coefficients will be different. The coefficients are time-dependent. From comparing these two equations, we can immediately see that c_n of t is the original coefficients for this initial state times the phase factor. Note here that the time evolution of energy eigenstate is given by a simple phase factor. In other words, the quantum state does not change. This must also be true for any eigenstates of an operator that commute with Hamiltonian because if an operator commute with Hamiltonian, then that operator shares the same set of eigenvectors or eigenket with the Hamiltonian. These eigenstates are also eigenstates of Hamiltonian, and therefore their time evolution is given by a simple phase factor as well, meaning that their quantum state doesn't change either. These eigenstates that don't change over time is called constant of motion in analogy to classical physics. If we can find a complete set of operators that commute with Hamiltonian, then we can express on arbitrary initial state as a linear superposition of these eigenstates because those eigenstates form a complete set. The time evolution of this arbitrary initial state is then expressed by simply multiplying a phase factor E to the negative iE_nt over h-bar to each of the eigenstate in that superposition. This is analogous to the conserved quantity in classical physics. For example, consider momentum conservation in classical physics. In quantum mechanics, the similar situation would be a problem where momentum operators commute with Hamiltonian. Then the momentum eigenstate would be an eigenstate of Hamiltonian as well, and therefore the time evolution of momentum eigenstate can be described by multiplication of this simple phase factor, and therefore they don't change, the quantum state doesn't change over time. This is equivalent to two momentum conservation in classical physics. Now, let's consider expectation value. Consider an operator A which commute with Hamiltonian, and operator B which does not. Then calculate the expectation value of B with the initial state of ket a, which is an eigenstate of operator A. By definition, the expectation value of B with this initial state a is given by this. The expectation value is taken with a state evolved in time so that there is a time t here. But the time evolution of the state a, because it is an eigenstate of Hamiltonian, is given by this phase factor. This phase factor, you add it to the ket, and this phase factor, complex conjugate of the other phase factor attached to the bra. They cancel each other out. Remember, these are just numbers so they commute with any operator, so you multiply this and this first, cancel it out, and that gives you this equation here, which is the expectation value of B taken at t equals 0. Which means that the expectation value of B doesn't change. Why? Because we should expect this because the ket itself is a stationary state, it doesn't change over time. Why should the expectation value change? If the initial state is a superposition state, let's say it's an Alpha. An Alpha is some superposition state of the eigen ket A with a coefficient c sub a prime. Then if you evaluate the expectation value of B, you can write like this. Then you will have this double summation that has this exponential factor, time-dependence factor, with a frequency of oscillation. These are all sine and cosine oscillatory term. The frequency of oscillation is given by this energy difference here in the numerator divided by h bar. That will be the oscillatory term. What you should expect here is if the initial state is not an eigenstate of a Hamiltonian, and therefore, is not a stationary state, the expectation value will change over time and the specific time dependence of those expectation value will contain a series of oscillatory term, which oscillates with a frequency corresponding to the energy difference involved in the eigenstates in the series expansion or superposition expression. Now let's consider a specific example of electrons spin in a magnetic field. The Hamiltonian for a spin magnetic field interaction is given by this. This S here is the spin angular momentum of an electron, and the B here is the external magnetic field that you apply. If the magnetic field is static and pointed along the z direction, then this dot-product simply gives you the z component of the spin and b here is the magnitude of your applied magnetic field. Now, with this expression, it's obvious that the Hamiltonian, H, and the spin angular momentum commute because Hamiltonian is simply given by multiplying a constant to the z component of spin. The eigenstate of z, which we know there are two; spin up and spin down, and the z plus and z minus we denote them with. The eigenvalues for the z plus and z minus eigenstates would be plus h bar over 2 and minus h bar over 2 as shown here. These eigenstate, z plus, and z minus of operator as c is also an eigenstate of Hamiltonian H, and the energy eigenvalue, from this definition, falls out immediately like this. For convenience, we define parameter Omega as shown here. If we do that, we can write down the Hamiltonian operator as a simple product of the S sub z operator, the z component to z operator, multiplied by this scalar parameter Omega. Because the Hamiltonian is a scalar multiple of S sub z, the eigenkets of S sub z operator are also eigenkets of Hamiltonian. The eigenvalues are now different because the Hamiltonian has this scalar multiple Omega, the energy eigenvalues are now plus n minus h bar omega over 2, instead of simply plus n minus h bar over 2, in the case of S sub z. This establishes the fact that z plus and z minus, the eigenkets of spin z, S sub z operator, are also eigenstates of the Hamiltonian. This is very important because the energy eigenkets, the time evolution of the energy eigenkets, is given by simply multiplying the exponential phase factor, as we have discussed earlier. Let's consider an arbitrary ket spin state, and we can express them as a superposition of Z plus and Z minus, using Z plus and Z minus as the basis set, the C plus and C minus are the complex numbers coefficients, and the time evolution of this arbitrary ket Alpha is then given by simply multiplying these exponential phase factor to Z plus and these exponential phase factors to Z minus, because once again, Z plus and Z minus are energy eigenstates and their time evolution is given by this exponential phase vectors. Now, let's consider a special case of when the initial state, this Alpha_t initial state happens to be X plus the spin upstate along the X-direction. Now, from the discussion of Stern-Gerlach experiments, we know that X plus can be expressed in terms of Z plus and Z minus as shown here, one over square root of two times Z plus, plus one over root two times Z minus. That's an expression for X plus and for X minus, all you need to do is simply change this plus sign in the middle to a minus sign. Now, the time evolution of these X plus state then is given by this equation here, all you need to do is to substitute one over root two, one over root two to for C plus and C minus respectively. Now, then what we can do is to find the probability of measuring spin up or spin down along the X direction as a function of time. How do we do that? We find the quotients by taking the inner product of this time about state Alpha T, which we found in the previous slide, with the eigenstates of x_z, X plus, or X minus. If you take this inner product, it gives you the coefficients, and if you take the absolute value square of the coefficients, that will give you the probability of measuring the respective state. What we need to do is to substitute this known expression of X plus and X minus in terms of Z plus and Z minus, and then, of course, the expression for Alpha_t, which we found in the previous slide. Doing some algebra, for measuring X plus spin up along X, you get cosine squared of Omega t divided by 2, the probability of measuring spin down along X as X minus, you get the probability evolves as sine squared of Omega t over 2. We can also calculate the expectation value of S_X, the X component of spin operator. Following the standard definition, what you need to do is to multiply the Alpha of t Bra from the left and then Alpha of t Ket from the right to this operator S_X. Doing some algebra, you will find that the expectation value is given by h bar over two times cosine Omega t, and you can do the same for the Y spin S_y, and then you will get h bar over two times sine Omega t, and for completeness, if you do S_z, you will get zero. What does this mean? When you're measuring spin X, Y, and Z component, Z component remains zero always, and the X component and Y component oscillate in time with frequency Omega, but there is a 90-degree phase difference. F_x is given by cosine function, S_y is given by sine function. What does that mean? That means your spin initially at T equals zero, it was the X plus pointing X plus direction, so you measure spin up along X, but as you evolve time, then the spin rotates about Z-axis as shown here. The spin remains in the XY plane, and that's what it means to be too heavy to have the expectation value of S_ Z0 here, and the X and Y component oscillates with Omega with 90-degree phase difference between the two represents a vector rotating in the XY plane. This phenomenon that we expect for a spin angular momentum for an electron in a constant magnetic field is called spin precession.
