Chapter 4: Time evolution

Note

Those who are not shocked when they first come across quantum theory cannot possibly have understood it. ― Niels Bohr, Essays 1932-1957 on Atomic Physics and Human Knowledge

Summary

Attention

So far, we have built a formal vector space populated by kets (and its equivalent space populated by bras). We have also introduced operators, which allow us to transform kets into other kets (those operators, to preserve norms,must be unitary). We also encountered other operators, including Hermitian ones, which correspond to observables (e.g., the spin angular momentum operators). All in all, we realized that the rules that govern the manipulation of those objects are the rules of linear algebra. Finally, we already established the importance of commutators and all the associated properties stemming from linear algebra. When we spoke about rotation of the spin angular momentum states, we used angles of rotation as parameters of our transformations. Now is time to introduce another example of unitary transformation, with time t used as the parameter of the transformation (for the record: time is not an operator!).

This chapter introduces the time evolution operator $\hat{U}(t)$ , which evolves a ket initially in state $\ket{\psi(0)}$ into a ket at time t: $\ket{\psi(t)}=\hat{U}(t)\ket{\psi(0)}$ . Formally, we used the exact same approach to define the operator of time translation for infinetisimal time $dt$ by introducing the Hermitian operator $\hat{H}$ in a way similar to the way we introduced the operator $\hat{J}_z$ as the generator of rotation around the $z$ axis (see Chapter 2: Rotation of basis states and matrix mechanics).

We realize that this formal introduction naturally leads to Schrodinger equation and that $\hat{H}$ is the Hamiltonian (i.e., the energy operator) of the system. By construction, this operator is Hermitian and its eigenvalues (which are real) are the energies $E$ , corresponding to the energy states $\ket{E}$ (formally: $\hat{H}\ket{E}=E\ket{E}$ ). Here, we see another instance of the use of a label to write the ket vector using information we know about these particular states (that is: we know what their energy is).

We also encounter functions of operators and we stress that much care has to be taken when trying to evaluate the effect of such a function on a quantum state vector (see box below for details). Specifically, we see that when the Hamiltonian does not depend explicitly on time, $\hat{U}(t)=\mathrm{e}^{-i\hat{H}t/\hbar}$ . In this case, when applied to an energy eigenstate of the Hamiltonian, the time evolution reduces to the introduction of an meaningless overall phase, and thus the system does not change with time, this is why we say that the energy state is a stationary state.

We also look into the time evolution of the expectation value of an operator and find a very important relationship involving a commutator (again!). The formula is provided below. Particularly if a time-independent operator commutes with the Hamiltonian, its expectation value is constant with time (in other words, it corresponds to a constant of motion).

We then explore two specific examples of time evolution in two-state problems. In the first one, we consider a magnetic field with a constant component along the z axis and an oscillating component along the x axis. The corresponding Hamiltonian involves both the angular momentum projection on x and z. As a result, the state $\ket{+z}$ is no longer a stationary state and the state of the system will evolve with time. The problem can be solved exactly (to lead to Rabi’s equation) but, in the course, we focus on a frequency range close to the resonance frequency. At that frequency, the spin system periodically emits and absorbs energy (the energy comes from the external field). Because the largest signal (i.e., resonance) occurs at $\omega=\omega_0=g e B_{0} / 2 m c$ , this allows for the determination of $\omega_0$ and thus of property of the material (this is why this technique is used in MRI or NMR – NMR because it is the spin-1/2 of the proton that’s used). If you want to learn more about MRI, visit this site.

Ammonia molecules ( $NH_3$ ) constitute another example of a 2-state system where each state corresponds to the position of the N atom either on top or under the plane defined by the three H atoms. We first show that these geometrical positions are not eigenstates of the system, due to the small (but non-zero) probability of the system to flip between the two states (special note to students: do not be confused, we are not considering the rotation of the molecules; imagine the plane of H atoms to be fixed during the transition). Here, the driving external field is an electric field since it can interact with the intrinsic electrical dipole moment of the ammonia molecules (which is due to the non-uniform distribution of charge around hydrogen and nitrogen, the latter featuring a lone electron pair). The mathematics of the problem is the same as that of the magnetic resonance problem and the resulting process involves a resonance where the molecule periodically absorbs and emits well defined lumps of energy, from and to the external field. Because that energy corresponds to radiations in the microwave frequency range, this system is called a MASER. Students will remember that the lumps of energy are associated with the physical process of oscillations between the two states in a 2-state problem.

Finally, we discuss that the uncertainty relationship $\Delta E \Delta t \geq \frac{\hbar}{2}$ is not of the same type as what we studied in chapter 3. Here, we remember that t is not an eigenvalue but rather a parameter. It is not appropriate to talk about uncertainties here, instead students will read this equation as a relationship between the typical time it takes for a system to transition between two states separated by $\Delta E$ . In a nutshell, the larger the energy difference, the smaller the time the system resides in a given state.

Note

Quick recap on operators

As a reminder a quantum operator (denoted with a $\hat{.}$ sign), operates on a ket, placed on its right to yield another ket.

$\hat{O}\ket{\psi}=\ket{\phi}$
If the operation preserves norm (e.g., rotation, time evolution), it is unitary:

$\hat{O}^{\dagger}\hat{O}=1$

In other words, the adjoint is the inverse of the operator (note: the “1” on the right-hand side of the equation above is actually an operator as well (the identity operator) but we do not conventionally place a $\hat{.}$ sign on it, by bad habit.
An adjoint operator operator operates on the left of a bra vector:

$\bra{\psi}\hat{O}^{\dagger}=\bra{\phi}$
Sometimes, we have a choice if an operator operates on the left or on the right!

For example

$\bra{E}\hat{H}\ket{E'}$

can be calculate in two equivalent ways. We can operate to the left (we can do this because the Hamiltonian operator is its self-adjoint, that is: Hermitian):

$\bra{E}\hat{H}\ket{E'}=E\bra{E}\ket{E'}=E\delta_{E,E'}$

or, by operating on the ket first:

$\bra{E}\hat{H}\ket{E'}=E'\bra{E}\ket{E'}=E'\delta_{E,E'}=E\delta_{E,E'}$

Warning

Effect of a function of an operator on a state

When we have a function of an operator, it is not straightforward to see how this operator (since the function of an operator is an operator) operates on a state vector. There are two ways to do this:

In the most general case, you do this by expanding the function of the operator in its Taylor series, for example:

$e^{-i \hat{H} t / \hbar}=\left[1-\frac{i \hat{H} t}{\hbar}+\frac{1}{2 !}\left(-\frac{i \hat{H} t}{\hbar}\right)^{2}+\cdots\right]$

Then we can apply the operators multiple times as needed.
There is a more convenient way to do this when you operate on an eigenstate of the operator. Using the example above, we have

$e^{-i \hat{H} t / \hbar}|E\rangle=e^{-i E t / \hbar}|E\rangle$

The difference between the left and the right-hand sides is striking: we have an operator on the left and a number on the right.
More generally, for a function $f(\hat{A})$ of an operator $\hat{A}$ whose eigen-solutions are $\hat{A}\ket{a_i}=a_i\ket{a_i}$ , we have the important property that:

$f(\hat{A})\ket{a_i}=f(a_i)\ket{a_i}$
Thanks to the previous result, we have a convenient way to operate on a general ket:
1. First you write the general ket using the basis of the eigenvectors of the operator like we did in Chapter 1:
  
  $\ket{\psi}=\sum_i c_i \ket{a_i}$
2. Then you can use the result described previously to write:
  
  $f(\hat{A})\ket{\psi}=\sum_i c_i f(a_i) \ket{a_i}$
Conclusions: Do not take shortcuts! It is easy to make a mistake if you do not pay attention!

Learning Material

Copy of Slides

The slides for Chapter 4 are available in pdf format here: 📂.

Screencast

Key Learning Points

The time translation operator is a unitary operator defined as $\hat{U}(t)\ket{\psi(0)}=\ket{\psi(t)}$
The infinitesimal time translation allows us to introduce the generator of time translation:

$\hat{U}(d t)=1-\frac{i}{\hbar} \hat{H} d t$
The generator of time translation $\hat{H}$ is the Hamiltonian of the system. This operator returns the energy of a system and it is Hermitian (since all outcome of it is a real number)
A simple manipulation indicates that the definitions and properties above lead to the time-dependent Schrodinger equation:

$i \hbar \frac{d\ket{\psi(t)}}{d t}=\hat{H}\ket{\psi(t)}$
For a Hamiltonian that does not explicitly depend on time, we also found that:

$\hat{U}(t)=\lim _{N \rightarrow \infty}\left[1-\frac{i}{\hbar} \hat{H}\left(\frac{t}{N}\right)\right]^{N}=e^{-i \hat{H} t / \hbar}$
Time evolution of the expectation value of an operator $\hat{A}$ is given by the important result:

$\dv{t} \expval{A}=\frac{i}{\hbar}\expval{[\hat{H}, \hat{A}]}{\psi(t)}+\expval{\frac{\partial \hat{A}}{\partial t}}{\psi(t)}$
The Hamiltonian of a spin-1/2 particle interacting with an external magnetic field aligned along z is

$\hat{H}=\omega_{0} \hat{S}_{z}$

where $\omega_{0}=g e B_{0} / 2 m c$
The Hamiltonian of a spin-1/2 particle interacting with a constant external magnetic field aligned along z and an oscillating external magnetic field aligned along x:

$\hat{H}=\omega_{0} \hat{S}_{z}+\omega_{1}(\cos \omega t) \hat{S}_{x}$
Rabi’s formula:

$|\langle-\mathbf{z} \mid \psi(t)\rangle|^{2}=\frac{\omega_{1}^{2} / 4}{\left(\omega_{0}-\omega\right)^{2}+\omega_{1}^{2} / 4} \sin ^{2} \frac{\sqrt{\left(\omega_{0}-\omega\right)^{2}+\omega_{1}^{2} / 4}}{2} t$

Test your knowledge

Provide an expression of the operator of time translation for infinitesimal time dt.
Show that the generator of time translation is Hermitian.
Provide an expression of the time translation operator in the case when the generator of time translation is time independent.
Suppose $\ket{E}$ is an eigenstate of the generator of time translation, with eigenvalue $E$ . How does this state evolve with time? (you can use result from question 1 or question 3).
For a time independent generator of time translation, the expectation value of the energy of a system is conserved. True or False?
An eigenstate of an Hamiltonian at time $t=0$ is a stationary state. Is this always true? Why?
Is the time evolution operator Hermitian or Unitary? A. Hermitian B. Unitary C. Both D. Neither
What is the generator of the time evolution operator? A. the parameter $t$ B. the Hamiltonian C. the Lagrangian D. none of the above
We have an operator $\hat{A}$ that (1) commute with the Hamiltonian and (2) has no explicit time dependence. What can you conclude? A. The expectation value of that operator for any state is constant. B. The expectation value of that operator is constant, but only if the state if an energy eigenstate. C. One can’t say much about the expectation value but we know the operator $\hat{A}$ and $\hat{H}$ share eigenvalues.
What’s the main idea behind magnetic resonance? A. A spin 1/2 rotates when subjected to an external magnetic field oriented along $z$ . B. A spin 1/2 dynamically changes orientation when subjected to an external field oriented along $z$ . C. A spin 1/2 dynamically changes orientation when subjected to an external field oriented along $z$ with a time-dependent component of small amplitude along $x$ .

Hint

Find the answer keys on this page: Answers to selected test your knowledge questions. Don’t cheat! Try solving the problems on your own first!

Homework Assignment

Solve the following problems from the textbook: 4.5,4.10,4,11,4,15

Recitation Assignment

Solve the following problems from the textbook: 4.4, 4.10, and 4.13