Chapter 2: Rotation of basis states and matrix mechanics

Note

Quantum theory provides us with a striking illustration of the fact that we can fully understand a connection though we can only speak of it in images and parables. ― Werner Heisenberg

Summary

Attention

In Chapter 1: Stern-Gerlach Experiments, we introduced the ket vector ( $\ket{\psi}$ ). In this chapter, we look into how one can transform a state vector. This leads us to introduce the concept of operators. We also introduce the concept of representations which is a way to use a complete basis to express bra, ket, and operators as a row vector, column vector, and square matrix, respectively.

This chapter also introduces the concept of rotation. A rotation of a state vector is an operation that changes the state vector without changing its norm (in other words, the operator has to be unitary, $\hat{A}^{-1}=\hat{A}^{\dagger}$ ).

We learn about matrix representations of operators and on how one can transform one representation in a given basis into a representation in a different basis. We also realize that the expectation value of an operator is really a matrix element of the operator. The concept of matrix elements in quantum mechanics is important, and will be explored many times in this course.

The formalism described thus far to explain the Stern-Gerlach experiment for the spin of an electron can be directly applied to any other two-state problems (and, in fact, will be generalized easily to other systems as well in this course). For example, if one considers light as a two-state system, where the polarization can take on one of two values, one can apply the formalism of rotation of basis vectors to find that, for light, the projection of the spin is actually an integer number. This important result, which points to a crucial distinction between different types of quantum particles (i.e.,fermion versus bosons, as we shall see later in this course, in Chapter 12: Identical Particles) emerges from the formalism described thus far. It would seem we are on the right path!

The importance of generators in quantum mechanics

We will see in this course a number of examples where we want to express a unitary transformation (here a rotation around an axis) in terms of a small change in a parameter (in this case a infinitesimal angle $d\phi$ ), using an expression like this:

$\hat{R}(d \phi \mathbf{k})=1-\frac{i}{\hbar} \hat{J}_{z} d \phi$

It is crucial to realize that this equation is a cornerstone of our entire approach to quantum mechanics. Students should take their time to look at it carefully, so to realize it “makes sense”.

Let’s dissect it:

The operator $\hat{J}_{z}$ generates the rotation about the $k$ axis.
Don’t let the presence of $i$ and $\hbar$ , disturb you! Since we have not defined what $\hat{J}_{z}$ is, consider their presence as an arbitrary choice (it is not arbitrary as we shall see later).
With this definition, a dimensional analysis indicates that $\hat{J}_{z}$ has the dimension of an angular momentum (like $\hbar$ ).
This form looks a bit like the first term of a Taylor series (all higher order terms disappear since we have an infinitesimal rotation angle).

The way the rotation operator is defined leads us to a very important conclusion: the generator of the rotation must be Hermitian if the rotation is unitary! This is a very important concept: this means that for any of this type of transformations, there is an observable, that is: a physical properties one can measure!

Here, we show in the course that $\hat{J}_{z}$ is actually the angular momentum of the particle.

Finally, the definition above is expanded to the general case of a finite angle rotation. We find that:

$\hat{R}(\phi \mathbf{k})=\lim _{N \rightarrow \infty}\left[1-\frac{i}{\hbar} \hat{J}_{z}\left(\frac{\phi}{N}\right)\right]^{N}=e^{-i \hat{J}_{z} \phi / \hbar}$

Be careful with that equation and the meaning of the exponential of an operator!!!! If one does not know the eigenvalues of the operator, we must rely on the use of a Taylor series to evaluate it.

Note

Hermitian vs. unitary

Main properties: a unitary operator preserves norms and a Hermitian operator has real eigenvalues.
Mathematically:
- $\hat{U}^{\dagger} \hat{U}=1$ for a unitary operator. It is also called a rotation operator (in the general sense of preserving norms)
- $\hat{H}^{\dagger}=\hat{H}$ for a Hermitian operator.
A Hermitian operator corresponds to an observable since, in nature, observations correspond to real numbers.
If two operators have the same spectrum (i.e., same eigenvalues), they only differ from each other by a unitary transformation. In other words, their matrix representations can coincide by a change of reference frame.

Learning Material

Copy of Slides

The slides for Chapter 2 are available in pdf format here: 📂.

Screencast

Key Learning Points

Attention

An operator ( $\hat{A}$ ) is a mathematical object to transform a ket vector into another.

$\ket{\psi}=\hat{A}\ket{\phi}$
We learned that if we have a complete basis ( $\{ \ket{+z},\ket{-z}\}$ ), we can write any state vector as a linear combination. The state is therefore completely described by the complex coefficients:

$|\psi\rangle=\underbrace{\langle+\mathbf{z} \mid \psi\rangle}_{c_{+}}|+\mathbf{z}\rangle+\underbrace{\langle-\mathbf{z} \mid \psi\rangle}_{c_{-}}|-\mathbf{z}\rangle$
We can therefore represent a ket vector as a numerical vector:

$|\psi\rangle \underset{s_{z} \text { basis }}{\longrightarrow}\left(\begin{array}{c}\langle+\mathbf{z} \mid \psi\rangle \\ \langle-\mathbf{z} \mid \psi\rangle\end{array}\right)=\left(\begin{array}{c}c_{+} \\ c_{-}\end{array}\right)$
Likewise, a bra vector can be represented as a row vector

$\langle\psi| \underset{s_{z} \text { basis }}{\longrightarrow}(\langle\psi \mid+\mathbf{z}\rangle,\langle\psi \mid-\mathbf{z}\rangle)$

It is important to remember that a row (bra) or column (ket) representation only makes sense if one knows what (complete) basis is used. We also remember that we can perform operations only between representations formulated in the same basis. For example, the dot product is

$\langle\psi \mid \psi\rangle=\underbrace{(\langle\psi \mid+\mathbf{z}\rangle,\langle\psi \mid-\mathbf{z}\rangle)}_{\text {bra vector }} \underbrace{\left(\begin{array}{c}\langle+\mathbf{z} \mid \psi\rangle \\ \langle-\mathbf{z} \mid \psi\rangle\end{array}\right)}_{\text {ket vector }}=1$
A rotation around an axis can be expressed in terms of a generator (necessarily Hermitian):

$\hat{R}(\phi \mathbf{k})==e^{-i \hat{J}_{z} \phi / \hbar}$
The eigenvalues and eigenvectors of the generator of rotations around the $z$ axis are:

$\hat{J}_{z}|\pm \mathbf{z}\rangle=\pm \frac{\hbar}{2}|\pm \mathbf{z}\rangle$
The identity and projection operators:
1. $\hat{P}_{+}=|+\mathbf{z}\rangle\langle+\mathbf{z}|$ is a projection operator for the SGz experiment, it corresponds to the state coming out from the “upward” channel.
2. $\hat{P}_{-}=|-\mathbf{z}\rangle\langle-\mathbf{z}|$ is a projection operator for the SGz experiment, it corresponds to the state coming out from the “downward” channel.
3. The sum of the two operators yield the identity operator $\hat{I}=|+\mathbf{z}\rangle\langle+\mathbf{z}|+|-\mathbf{z}\rangle\langle-\mathbf{z}|$ . It is a mathematical representation of the modified SG experiment introduced in Chapter 1.
The same way as a complete basis can be used to represent any state vector in a vector form, a basis can be employed to express an operator in a matrix form. In a two-state system with basis $\{\ket{\phi_1}, \ket{\phi_2}\}$ , the general matrix form of the matrix of an operator $\hat{A}$ is:

$\left(\begin{array}{cc} \bra{\phi_1}\hat{A}\ket{\phi_1} & \bra{\phi_1}\hat{A}\ket{\phi_2} \\ \bra{\phi_2}\hat{A}\ket{\phi_1} & \bra{\phi_2}\hat{A}\ket{\phi_2} \end{array}\right)$
The matrix representation of an adjoint operator is the transpose-conjugate of the representation of the operator itself.

$A_{i j}^{\dagger}=A_{j i}^{*}$
Vectors and matrices can be expressed in different bases and it is also possible to move from one basis to another. For example, if one wants to get the $\{\ket{+x},\ket{-x}\}$ representation of an operator given in the $\{\ket{+z},\ket{-z}\}$ basis, we define the unitary matrix:

$\mathbb{S}=\left(\begin{array}{cc} \langle+\mathbf{z} \mid+\mathbf{x}\rangle & \langle+\mathbf{z} \mid-\mathbf{x}\rangle \\ \langle-\mathbf{z} \mid+\mathbf{x}\rangle & \langle-\mathbf{z} \mid-\mathbf{x}\rangle \end{array}\right)$

to obtain the new representation as

$\begin{array}{l} \left(\begin{array}{ll} \langle+\mathbf{x}|\hat{A}|+\mathbf{x}\rangle & \langle+\mathbf{x}|\hat{A}|-\mathbf{x}\rangle \\ \langle-\mathbf{x}|\hat{A}|+\mathbf{x}\rangle & \langle-\mathbf{x}|\hat{A}|-\mathbf{x}\rangle \end{array}\right) =\mathbb{S}^{\dagger}\left(\begin{array}{ll} \langle+\mathbf{z}|\hat{A}|+\mathbf{z}\rangle & \langle+\mathbf{z}|\hat{A}|-\mathbf{z}\rangle \\ \langle-\mathbf{z}|\hat{A}|+\mathbf{z}\rangle & \langle-\mathbf{z}|\hat{A}|-\mathbf{z}\rangle \end{array}\right)\mathbb{S} \end{array}$

likewise, for a vector, we can move from the z-basis to the x-basis:

$\left(\begin{array}{c} \langle+\mathbf{x} \mid \psi\rangle \\ \langle-\mathbf{x} \mid \psi\rangle \end{array}\right)=\mathbb{S}^{\dagger} \left(\begin{array}{c} \langle+\mathbf{z} \mid \psi\rangle \\ \langle-\mathbf{z} \mid \psi\rangle\end{array}\right)$

This chapter allowed us to find a much easier formulation to calculate the expectation value of an operator for a given state:

$\expval{S_{z}}_{\ket{\psi}}=\expval{A}{\psi}$

The advantage of this expression is the ease to apply it when the states and operators are expressed in a given basis. Note that here, we explicitly stated that expectation value of an operator is defined for a given state vector. We also see that the expectation value has the form of a matrix element in the sense of what we defined in point 8. above.

Test your knowledge

An operator of rotation is always a unitary operator…
1. Yes, and its eigenvalues are all positive.
2. Yes, and its eigenvalues are all real.
3. Yes, and its eigenvalues are usually complex.
4. No, such an operator must be Hermitian.
5. There is not enough information to answer this question conclusively.
The generator of rotation is an Hermitian operator…
1. Yes, and its eigenvalues are all positive.
2. Yes, and its eigenvalues are all real.
3. Yes, and its eigenvalues are usually complex.
4. No, such an operator must be unitary.
5. There is not enough information to answer this question conclusively.
What are the eigenvalues of the generator of rotation for spin- $\frac{1}{2}$ along the $z$ -axis?
1. 0 and 1
2. $\pm \hbar/2$
3. $\pm \hbar$
4. ${\rm e}^{\pm i\phi/2}$
Unlike rotation operators in Cartesian space, a quantum mechanical operator operates in the quantum space. It must preserve the norm of the state, otherwise, it cannot possibly be a rotation.
1. True, and the norm of the vector is arbitrary.
2. True, and the norm of the vector is exactly one.
3. False, it is perfectly fine for a rotation operator to collapse the state.
4. False, the norm of a state vector can also be chosen arbitrarily.
Two state vectors differ only by an overall phase, therefore…
1. They describe the exact the same system.
2. They can describe any two different systems so long as there exists a rotation operator that transforms one into another.
3. One can’t conclude anything from the provided information.
Think about the projection operator $\hat{P}_+=\ket{+z}\bra{+z}$ relative to the $SG_z$ experiment. Is this operator Hermitian or unitary?
1. It is Hermitian.
2. It is unitary
3. It is neither Hermitian nor unitary.
Consider the following complete basis: $\{ \ket{+x},\ket{-x}\}$ . What can you say about this operator: $\hat{O}=\ket{+x}\bra{+x}+\ket{-x}\bra{-x}$ ?
1. This is a beautiful operator! I will print it in high resolution and make a poster of it for my dorm!
2. This operator is actually the identity operator, it is reminiscent of the modified Stern-Gerlach experiment (for magnetic field along the $x$ axis).
3. This is a projection operator that does not preserve the norm.
What is the representation of $\ket{\psi}=\sqrt{\frac{{2}}{{3}}}\ket{+z}-\sqrt{\frac{1}{{3}}}\ket{-z}$ using the $\{\ket{+z},\ket{-z}\}$ basis?
1. $(\sqrt{\frac{2}{{3}}},-\sqrt{\frac{1}{{3}}})$
2. $\left(\begin{array}{l}~\sqrt{\frac{{2}}{3}}\\-\sqrt{\frac{1}{3}}\end{array}\right)$
3. This is not a valid state vector!
The eigenvalues of the matrix representation of an operator do not depend on the representation used to write the matrix.
1. True, but only if the operator is Hermitian
2. True
3. False
4. That’s a very good question. I’m glad you asked.
You are given two operators $\hat{A}$ and $\hat{B}$ in matrix representations from two different books, but the bases used have not been provided to you. What can you do? (read carefully)
1. Multiply the two matrices in order to learn more about $\hat{A}\hat{B}$ .
2. Calculate the eigenvalues of each operator separately.
3. 1. and B. are correct.
4. None of the other answers is correct.
Using the $\{\ket{+x},\ket{-x}\}$ basis, what is the matrix representation of the $\hat{P}+=\ket{+z}\bra{+z}$ operator? (for this question, you can use the fact that $\ket{\pm x}=\frac{1}{\sqrt{2}}(\ket{+z}\pm\ket{-z})$ .
1. $\left(\begin{array}{ll}1 & 0 \\0 & 0\end{array}\right)$
2. $\left(\begin{array}{ll}1 & 1 \\0 & 0\end{array}\right)$
3. $\frac{1}{2}\left(\begin{array}{ll}1 & 1 \\1 & 1\end{array}\right)$
4. $\frac{1}{2}\left(\begin{array}{rr}1 & -1 \\-1 & 1\end{array}\right)$
Using the $\{\ket{+y},\ket{-y}\}$ basis, what is the matrix representation of the $\hat{P}_-=\ket{-z}\bra{-z}$ operator? (for this question, you can use the fact that $\ket{\pm y}=\frac{1}{\sqrt{2}}(\ket{+z}\pm i\ket{-z})$ . Be careful about the bra version of a ket that has a complex number as a coefficient! Is the matrix Hermitian?
1. $\left(\begin{array}{ll}1 & 0 \\0 & 0\end{array}\right)$
2. $\left(\begin{array}{ll}1 & 1 \\0 & 0\end{array}\right)$
3. $\frac{1}{2}\left(\begin{array}{ll}1 & 1 \\1 & 1\end{array}\right)$
4. $\frac{1}{2}\left(\begin{array}{rr}1 & -1 \\-1 & 1\end{array}\right)$
5. $\frac{1}{2}\left(\begin{array}{rr}1 & i \\ i & 1\end{array}\right)$

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: 2.2; 2.5; 2.6; 2.11; 2.23

Recitation Assignment

Solve the following problems from the textbook: 2.3; 2.10; and 2.24

Recording of the live recitation session can be found here.