ECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline: Time-Dependent Schrödinger Equation Solutions to thetime-dependent Schrödinger Equation Expansion of Energy Eigenstates
Things you should know when you leave Key Questions How can we understand the time-dependent Schrödinger s equation? How do the solutions to change with time? How do our canonical problems change with time-dependence? M. J. Gilbert ECE 487 Lecture 6 02/03/11
Thus far we have assumed that most things were steady in time, but that was pretty unsatisfying because intuitively we know the solution contains motion with time evolution Consider several situations we have already encountered: 1. Simple harmonic oscillator 2. Electrons in an electric field So to understand these things we need to keep the time dependence in the Schrödinger equation. But remember this is still very different from the normal classical time dependent wave equation. To solve this problem, we need to introduce a very important concept in quantum mechanics, superposition states. These states allow us to handle the time evolution in quantum mechanics very easily.
The key to understanding the time dependence and the Schrödinger equation is understanding the relationship between frequency and energy in quantum mechanics A good example of this is the case of electromagnetic waves and photons. Imagine two experiments with a monochromatic electromagnetic wave. In one experiment, we measure the frequency of the oscillation in the wave. In a second experiment, we count the number of photons per second. So, we can count how many photons per second correspond to a particular power at this frequency However, this discussion is for photons and these particles are not well described by the Schrödinger equation.
But there is a problem -q +q Hydrogen atoms emit photons as they transition between energy levels. We expect some oscillation in the electrons at the corresponding frequency during the emission of the photon. E H = 4 0q 13. 6 m ev = 2 n 2 2 ( 4πε n) 0 n = 1, 2, 3, So we should also expect a similar relation between energy and frequency associated with the electron levels.
Keep in mind that at the end we still want to have a wave equation But it must be sensitive to the relationship between energy and frequency: We can guess a form for the solution to the wave equation with time dependence from our knowledge of electromagnetics, like when we postulated the Schrödinger equation in a uniform potential Let s postulate the form of the time-dependent Schrödinger equation to be of the form: Again, we can make this postulate in some way because we already know the answer. So don t be discouraged if you don t see how we can arrive at this equation yet.
The solution to such an equation should have waves in the form: With relations for the energy and wavevector which encorporate what we know about the time-independent form and what we know to be true about transitions: Which are indeed solutions to the postulated version of the Schrödinger equation we had on the previous slide so long as the potential is zero everywhere. Why is there a negative sign on the right hand side of the Schrödinger equation? So that a wave with a positive spatial part is definitely a wave propagating in the positive z-direction for all positive energies. A wave of the form:
How does this time-dependent equation compare with other timedependent equations that we have encountered previously? A more common classical wave equation has a different form: This equation also has some very familiar solutions: Some things to note thus far The wave equation above has a second order time-dependence whereas the Schrödinger equation only has one time derivative. The Schrödinger equation uses complex notation which implies that the wavefunction is required to be a complex entity. This is not like the use of complex notation in terms of a classical wave. We cannot take the real part of the calculated wave and compare it to a classical wave. M. J. Gilbert ECE 487 Lecture 6 02/03/11
To this point, we have just postulated the solution to the Schrödinger equation based on knowledge of the answer, now it is important to understand how the time-dependent solution relates to the timeindependent solution. Suppose we had a wave where the spatial variation of the wave did not change as a function of time Now multiply the wavefunction by a time varying factor, A(t), in front of the spatial part of the wavefunction: Solutions whose spatial behavior does not vary with time should satisfy the Schrödinger equation s time-independent form the addition of the time-varying factor does not change this: Form of A(t) doesn t matter
So, let us now substitute the full form of our time-dependent wavefunction into the time-dependent form of the Schrödinger equation. Remember we have assumed that there is no spatial dependence, therefore we can write down the energy eigenvalues: Or assuming some constant A 0 : Hence, if the spatial part is constant in time the full time-dependent wavefunction can be written as: The wavefunction is stable, but the probability density is not: M. J. Gilbert ECE 487 Lecture 6 02/03/11
Now that we are more comfortable with the time-dependence of the Schrödinger equation, we can start to think about solutions: The time-dependent Schrödinger equation is not an eigenvalue equation! It is not an equation that has solutions only for a particular set of values of a given parameter. Instead it allows us to calculate what happens as a function of time if we knew the wavefunction at each point in space at some time, t 0. Then we can evaluate the left hand side of the equation at that time for all r. We would know: And we can integrate to get: For all times.
To be more explicit, we would have: Because the Schrödinger equation tells us the derivative of the wavefunction as a function of time at time t 0, we know the wavefunction everywhere at time t 0. And we know how to calculate: In other words, the whole subsequent evolution of the wavefunction could be deduced from its spatial form at some given time. We could view this ability to deduce the wavefunction at all future times as the reason why this equation has a first derivative in time. As opposed to the classical second derivative. The spatial function sets the subsequent time evolution as if the spatial wavefunction is an eigenstate there is no variation other than oscillations. M. J. Gilbert ECE 487 Lecture 6 02/03/11
We already know that linearity helped a lot with the time-independent Schrödinger equation, how does it help in the time-dependent equation? The time-dependent Schrödinger equation is still linear in the wavefunction. No higher powers of ψ appear anywhere in the equation. If ψ is a solution, then so also is Aψ where A is any constant. Another consequence of linearity is the linear superposition of solutions. If Ψ a (r,t) is a solution and Ψ b (r,t) is also a solution the so also is Ψ a+b (r,t) = Ψ a (r,t) + Ψ b (r,t). This can be verified by substitution. We can also multiply the individual solutions by constants and still have a valid solution: Ψ c (r,t) = c a Ψ a (r,t) + c b Ψ b (r,t) where c a and c b are complex constants.
This is yet another concept which is not amenable to classical thoughts In classical mechanics a particle simply has some state associated with its particular position and momentum. Now we say a particle can exist in a superposition of states each of which may have different energies, positions, or momenta. Such superpositions are actually necessary in quantum mechanics so we can recover the behavior that we expect classically from particles. Let s now consider a way to look at the evolution of the wavefunction in the simplest case, when V is constant in time. We expand the wavefunction in the energy eigenfunction basis. Each eigenstate is a separate solution to the time-dependent Schrödinger equation. We already know that we can write the n-th eigenfunction as: M. J. Gilbert ECE 487 Lecture 6 02/03/11
Because of linearity, we know that the sum of any such solutions will also be a solution. Hence, the usefulness of linear superpositions in the timedependent Schrödinger equation. Suppose we want to expand the original spatial solution at time t = 0 in energy eigenfunctions: Where the a n are the expansion coefficients, which are generally complex. Remember that we can expand any spatial function like this because of completeness. We can now write down the corresponding time-dependent function: We already know that this is a solution. We can check by plugging in t = 0 and we get back our original wavefunction. M. J. Gilbert ECE 487 Lecture 6 02/03/11
Alright, we know one solution to the time-dependent Schrödinger equation expanded in the eigenenergy basis As long as there is no time-dependence in the potential V(r). With the initial condition: Therefore, if we expand the spatial wavefunction in the energy eigenstates at t = 0, then we have solved for the time evolution of the state hereafter No further integrating to do. We still need to perform a sum at each time of interest. M. J. Gilbert ECE 487 Lecture 6 02/03/11
Now let s look at a few simple cases where time is allowed to evolve and the potential is fixed in time with the system in a superposition state In particular, we want to reexamine two situations: 1. Simple linear superposition in an infinite quantum well 2. Harmonic Oscillator So, let s begin by remembering what we already know about the infinite quantum well Energies are quantized and there is no zero point energy Wavefunctions have even and odd partiy Solutions are standing waves and not travelling waves.
We are in an infinite quantum well and that the particle is in a normalized linear superposition state. We can write down a wavefunction which is equal parts of the first and second states of the well: From here, we can easily find the probability density: Things to notice: The probability density has a part that oscillates at an angular frequency: We could have added an arbitrary amount onto both of the energies E1 and E2 and it would not have changed the oscillation nature.
The solutions look like this Unit of time is taken to be ħ / E1. The oscillation angular frequency ω21 is 3 per unit time. Probability oscillates back and forth 3 times in 2π units of time. M. J. Gilbert ECE 487 Lecture 6 02/03/11
Now let s examine the harmonic oscillator example we introduced two lectures ago We can similarly construct a linear superposition state for the harmonic oscillator to now see the time dependent behavior. As before, let s construct a state which is a superposition of equal parts of the first and second states. In general if we make a linear combination of two energy eigenstates with energies E a and E b then With wavefunction: And probability density:
Look at the resultant superposition of the first two states:
The linear superpositions that correspond best to our classical understanding of a harmonic oscillator are known as coherent states. The coherent state for a harmonic oscillator of frequency ω is: where Are the harmonic oscillator eigenstates. Also notice that: Is the Poisson distribution from statistics with mean N and square root of N standard deviation. Therefore, we can calculate the probability density just by summing an appropriate number of terms.
Harmonic oscillator in a coherent state for N =1:
Harmonic oscillator in a coherent state for N =10:
Harmonic oscillator in a coherent state for N =100:
When the harmonic oscillator is in a coherent state: The probability distribution essentially oscillates back and forth from one side of the potential to the other with some angular frequency. It retains its shape as it moves. For higher N, the spatial width of the probability distribution function becomes a smaller fraction of the oscillation amplitude and appears very localized relative to the size of the oscillation. Just like we would expect classically. In general, a system in a linear superposition of multiple energy eigenstates does not execute a simple harmonic motion. The harmonic motion is a special consequence of the fact that all of the energy levels are equally spaced. M. J. Gilbert ECE 487 Lecture 6 02/03/11
To see what happens when the levels are not spaced evenly, consider the finite quantum well case: