• February 28th, 2025

Postulate II & III – Observable operators and eigenvalues

  • Author: Quentin Stern
  • Editor: Joshua Straub

The full scripts presented in this post are available at these links: MATLAB, Python

With the first postulate of quantum mechanics, we learnt that everything there is to know about a quantum system is encapsulated in the wave function \ket{\psi}. But so far, the wave function is sort of a black box and we still need to extract relevant information from it. The second and third postulates tell us how to extract measurable quantities from the wave function. We will look at both postulates at the same time, as it’s easier to make sense of them together.

Formulation of the postulates

The second postulate defines the mathematical form of measurable quantities, also known as observables.

Postulate II: Every physical observable A is associated with a Hermitian operator \hat{A} acting on \mathbb{H}. The eigenvectors of \hat{A} form a basis \mathbb{B} of \mathbb{H}.

The third postulate defines the possible outcome of a measurement of A.

Postulate III: Measuring the observable A can only result in one of the eigenvalues of \hat{A}. Because \hat{A} is Hermitian, its eigenvalues are real.

Let’s say that the system is a spin in a magnetic field and I want to measure its angular momentum along the z axis, I_z. The second postulate says that this measurable quantity is associated with an Hermitian operator that we denote \hat{I}_z, where the hat represents the fact that this object is an operator acting on the Hilbert space. The third postulate adds that the result of the measurement of \hat{I}_z can only be an eigenvalue of \hat{I}_z.

Operators and their eigenvalues

First, what is an operator? An operator acts on a Hilbert space by returning an element of the Hilbert space \mathbb{H} for any element of the same Hilbert space (that is, its action defines an endomorphism on \mathbb{H}). In the matrix representation of quantum mechanics, states (kets) are represented by vectors and operators by square matrices. When the state ket is multiplied with the operator matrix, another state ket comes out. We will see in a moment what this physically means.

The second postulate states that operators associated with observables are Hermitian, which has important consequences. Indeed, the spectral theorem states that for any Hermitian operator \hat{A}, there exists an orthonormal basis of \mathbb{H} formed by eigenvectors \ket{n} of \hat{A}. The fact that \ket{n} is an eigenvector of \hat{A} means that there exists a number a_n, called the eigenvalue, that satisfies the eigenequation,

(1)   \begin{equation*} \hat{A}\ket{n}=a_n\ket{n}.\end{equation*}

By the third postulate, the set of eigenvalues a_n of \hat{A} are the only possible outcomes of the measurement of \hat{A}. As stated in the third postulate, this eigenvalue is real because \hat{A} is Hermitian. That an operator is Hermitian means that it’s equal to its adjoint (i.e. \hat{A}^*=\hat{A}), which for a matrix, means that it is equal to its complex transpose. A consequence of this is that, despite the fact that the mathematical formulation of quantum mechanics involves complex numbers, measurements we perform in the lab still result in real numbers, as in classical physics.

With a bit of linear algebra, one can show that, because the eigenvectors of \hat{A} being a basis of \mathbb{H}, the matrix representation of \hat{A} can be written as

(2)   \begin{equation*} \hat{A}=\sum_{n}a_{n}\ket{n}\bra{n}, \end{equation*}

where the coefficients a_n are the eigenvalues of \hat{A} corresponding to the basis eigenvectors \ket{n}. This property will be used below. That the eigenvectors \ket{n} of \hat{A} form a basis of \mathbb{H} also implies that any state \ket{\psi} can be written as a function of \ket{n}

(3)   \begin{equation*} \ket{\psi}=\sum_{n}c_{n}\ket{n}, \end{equation*}

where c_k are normalized complex coefficients, i.e., \vert\braket{\psi\vert\psi}\vert^2=1.

Single spin in a magnetic field

Let’s come back to the measurement of the angular momentum along the z-axis of an isolated spin 1/2. We have mentioned in the previous post that the two possible values are +\hbar/2 and -\hbar/2. By the third postulate, these values are the eigenvalues of \hat{I}_z. We, therefore, have the following relations

(4)   \begin{equation*} \begin{array}{l} \hat{I}_z\ket{\alpha}= +\frac{\hbar}{2}\ket{\alpha}\\ \hat{I}_z\ket{\beta}=-\frac{\hbar}{2}\ket{\beta} \end{array}. \end{equation*}

The matrix representation for the operator \hat{I}_z in the Zeeman basis, which satisfies the above relations is

(5)   \begin{equation*}\hat{I}_z=\frac{\hbar}{2}\begin{pmatrix}+1 & 0 \\0 & -1 \end{pmatrix},\end{equation*}

which can be constructed using Eq. 2.

The matrix representation of the angular momentum operators along the two other orthogonal axes are

(6)   \begin{equation*}\hat{I}_x=\frac{\hbar}{2}\begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix},\end{equation*}

and

(7)   \begin{equation*}\hat{I}_y=i\frac{\hbar}{2}\begin{pmatrix}0 & -1 \\+1 & 0\end{pmatrix},\end{equation*}

which we give without proof. The three matrices used to define the angular momentum operators are called the Pauli matrices.

The three operators \hat{I}_x, \hat{I}_y, and \hat{I}_z that we have defined are the basic building blocks for the representation of any spin-1/2 system. To make our life easier down the line, we redefine these operators setting \hbar=1, not to carry this heavy factor all along. We’ll see later that this corresponds to expressing energy in terms of angular units (rad.s-1) rather than in terms of Joules. We therefore write the angular momentum operators as

(8)   \begin{align*} \hat{I}_x &=\frac{1}{2}\begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}\\ \hat{I}_y &=\frac{i}{2}\begin{pmatrix}0 & -1 \\+1 & 0\end{pmatrix}. \\ \hat{I}_z &=\frac{1}{2}\begin{pmatrix}+1 & 0 \\0 & -1 \end{pmatrix}\end{align*}

This affects their eigenvalues, which are no longer \pm\hbar/2 but now \pm1/2. These operators can be constructed in a computer program using standard matrix notation.

  • MATLAB
  • Python
  • Mathematica
% Definition of the angular momentum operators
Ix = [0  +1;+1  0]/2;
Iy = [0  -1;+1  0]*1i/2;
Iz = [+1  0; 0 -1]/2;
# Required package for numerical calculations
import numpy as np

# Definition of the angular momentum operators
Ix = np.array([[0,  +1/2],[ +1/2, 0]])
Iy = np.array([[0, -1j/2],[+1j/2, 0]])
Iz = np.array([[+1/2,  0],[0,  -1/2]])
none

Eigenequations of angular momentum operators

Eq. 4 shows the eigenequations for the \hat{I}_z operator, where the angular momentum expressed in SI units. Setting \hbar=1, we then have

(9)   \begin{equation*} \begin{array}{l} \hat{I}_z\ket{\alpha}= +\frac{1}{2}\ket{\alpha}\\ \hat{I}_z\ket{\beta}=-\frac{1}{2}\ket{\beta} \end{array}. \end{equation*}

To the contrary, \ket{\alpha} and \ket{\beta} are not eigenvectors of \hat{I}_x and \hat{I}_y. Indeed, we have for example,

(10)   \begin{equation*} \begin{array}{l} \hat{I}_x\ket{\alpha}= +\frac{1}{2}\ket{\beta}\\ \hat{I}_x\ket{\beta}=+\frac{1}{2}\ket{\alpha} \end{array}, \end{equation*}

which are not eigenquations. The eigenequations for the \hat{I}_x and \hat{I}_y operators can be found by diagonalizing their matrix representations and yield

(11)   \begin{equation*}\begin{array}{l} \hat{I}_x \left( \frac{\ket{\alpha}+\ket{\beta}}{\sqrt{2}} \right) = +\frac{1}{2}\left( \frac{\ket{\alpha}+\ket{\beta}}{\sqrt{2}} \right) \\ \hat{I}_x \left( \frac{\ket{\alpha}-\ket{\beta}}{\sqrt{2}} \right) = -\frac{1}{2}\left( \frac{\ket{\alpha}-\ket{\beta}}{\sqrt{2}} \right)\end{array},\end{equation*}

and

(12)   \begin{equation*}\begin{array}{l} \hat{I}_y \left( \frac{\ket{\alpha}+i\ket{\beta}}{\sqrt{2}} \right) = +\frac{1}{2}\left( \frac{\ket{\alpha}+i\ket{\beta}}{\sqrt{2}} \right) \\ \hat{I}_y \left( \frac{\ket{\alpha}-i\ket{\beta}}{\sqrt{2}} \right) = -\frac{1}{2}\left( \frac{\ket{\alpha}-i\ket{\beta}}{\sqrt{2}} \right)\end{array}.\end{equation*}

These eigenstate are sometimes defined using the following notation

(13)   \begin{equation*}\begin{split} \ket{\pm x} &= \frac{\ket{\alpha}\pm\ket{\beta}}{\sqrt{2}} \\ \ket{\pm y} &= \frac{\ket{\alpha}\pm i\ket{\beta}}{\sqrt{2}}, \end{split}\end{equation*}

because these two states correspond to the spin having its angular momentum along the x and y axes, respectively, as we will show in the next post. The factor \sqrt{2} in the denominator ensures that the states are normalized.

Conclusion

Now what does this all mean? What does it mean that a state that contains the sum or the difference of \ket{\alpha} and \ket{\beta}? These states are called superpositions. They are at the origin of fascinating quantum mechanical phenomena like entanglement. When the system is in state \ket{x}, it is neither in the \ket{\alpha} nor in the \ket{\beta} but in a superposition of the two. This leaves us with an open question: if we have several values that can result from the measurement of same physical system, how does Nature pick which one we’ll find? We can’t answer this deep question, but the next postulate will at least tell what the probability for each measurement will be.

To previous post:
Postulate I
To next post:
Postulate IV
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