Operator Algebra

In quantum mechanics, observable physical quantities like position, momentum, and energy are represented by mathematical operators. These operators act on the wave function (or state vector) of a system to yield information about these quantities. Operator algebra deals with the rules and properties governing these mathematical entities.

An operator, denoted by a caret symbol (e.g., $\hat{A}$), is a function that transforms one function into another. For instance, if $\psi(x)$ is a wave function, then $\hat{A}\psi(x) = \phi(x)$, where $\phi(x)$ is another function.

Types of Operators

Linear Operators

An operator $\hat{A}$ is linear if it satisfies the following conditions for any two wave functions $\psi_1$ and $\psi_2$, and any complex number $c$:

• $\hat{A}(\psi_1 + \psi_2) = \hat{A}\psi_1 + \hat{A}\psi_2$ (Additivity)
• $\hat{A}(c\psi_1) = c\hat{A}\psi_1$ (Homogeneity)

Most operators used in quantum mechanics are linear.

Hermitian Operators

Hermitian operators are crucial in quantum mechanics because they correspond to observable physical quantities, and their eigenvalues are always real. An operator $\hat{A}$ is Hermitian if it satisfies the following condition for any two wave functions $\psi_1$ and $\psi_2$ within the domain of the operator:

$$ \int_{-\infty}^{\infty} \psi_1^* (\hat{A}\psi_2) dx = \int_{-\infty}^{\infty} (\hat{A}\psi_1)^* \psi_2 dx $$

This can also be written using Dirac notation as:

$$ \langle \psi_1 | \hat{A} \psi_2 \rangle = \langle \hat{A} \psi_1 | \psi_2 \rangle $$

The integral is over all relevant space. For a position-space wave function $\psi(x)$, the integral is typically from $-\infty$ to $\infty$.

Operator Algebra Rules

Commutation Relations

The order in which operators act on a wave function can matter. The commutator of two operators $\hat{A}$ and $\hat{B}$ is defined as:

$$ [\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A} $$

If $[\hat{A}, \hat{B}] = 0$, then the operators $\hat{A}$ and $\hat{B}$ commute, meaning they can be applied in any order without changing the result. If $[\hat{A}, \hat{B}] \neq 0$, they do not commute.

Example: Position and Momentum Operators in 1D Let the position operator be $\hat{x}$ and the momentum operator be $\hat{p}_x = -i\hbar \frac{\partial}{\partial x}$. Let's calculate their commutator:

$$ [\hat{x}, \hat{p}_x] = \hat{x}\hat{p}_x - \hat{p}_x\hat{x} $$

Now, let's see how this acts on an arbitrary wave function $\psi(x)$:

$$ (\hat{x}\hat{p}_x - \hat{p}_x\hat{x})\psi(x) = \hat{x}(\hat{p}_x\psi(x)) - \hat{p}_x(\hat{x}\psi(x)) $$

$$ = \hat{x}(-i\hbar \frac{\partial}{\partial x}\psi(x)) - (-i\hbar \frac{\partial}{\partial x})(x\psi(x)) $$

$$ = -i\hbar x \frac{\partial\psi}{\partial x} + i\hbar (\frac{\partial}{\partial x}(x\psi(x))) $$

Using the product rule for differentiation: $\frac{\partial}{\partial x}(x\psi(x)) = 1\cdot\psi(x) + x\frac{\partial\psi}{\partial x}$.

$$ = -i\hbar x \frac{\partial\psi}{\partial x} + i\hbar (\psi(x) + x\frac{\partial\psi}{\partial x}) $$

$$ = -i\hbar x \frac{\partial\psi}{\partial x} + i\hbar \psi(x) + i\hbar x \frac{\partial\psi}{\partial x} $$

$$ = i\hbar \psi(x) $$

Since this holds for any $\psi(x)$, we have:

$$ [\hat{x}, \hat{p}_x] = i\hbar $$

This non-zero commutator is a fundamental result in quantum mechanics and implies that position and momentum cannot be simultaneously known with perfect accuracy (Heisenberg Uncertainty Principle).

Operator Addition and Multiplication

Operators can be added and multiplied similarly to numbers, provided they are linear.

• Addition: $\hat{A} + \hat{B}$ results in a new operator. For a wave function $\psi$, $(\hat{A} + \hat{B})\psi = \hat{A}\psi + \hat{B}\psi$.
• Multiplication: $\hat{A}\hat{B}$ means applying $\hat{B}$ first, then $\hat{A}$. For a wave function $\psi$, $(\hat{A}\hat{B})\psi = \hat{A}(\hat{B}\psi)$.

Operator multiplication is generally not commutative, i.e., $\hat{A}\hat{B} \neq \hat{B}\hat{A}$.

Hamiltonian Operator

The Hamiltonian operator, denoted by $\hat{H}$, is the operator corresponding to the total energy of a system. In classical mechanics, the Hamiltonian function $H$ is the sum of the kinetic energy $T$ and potential energy $V$. In quantum mechanics, we replace the classical variables with their corresponding operators.

For a single particle of mass $m$ in one dimension with potential energy $V(x)$, the Hamiltonian operator is:

$$ \hat{H} = \hat{T} + \hat{V} $$

Where:

• Kinetic Energy Operator ($\hat{T}$): $\hat{T} = \frac{\hat{p}_x^2}{2m} = \frac{1}{2m} (-i\hbar \frac{\partial}{\partial x})^2 = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}$
• Potential Energy Operator ($\hat{V}$): $\hat{V} = V(x)$ (For a simple potential, the operator is just multiplication by the potential function)

Therefore, in one dimension:

$$ \hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) $$

For a system of $N$ particles in three dimensions, the Hamiltonian operator becomes more complex, involving the sum of kinetic energies of all particles and the sum of all potential energy terms (particle-particle interactions, particle-external field interactions, etc.).

$$ \hat{H} = \sum_{i=1}^{N} \frac{\hat{p}_i^2}{2m_i} + V(\mathbf{r}_1, \mathbf{r}_2, ..., \mathbf{r}_N) $$

Where $\hat{p}_i$ is the momentum operator for the $i$-th particle and $\mathbf{r}_i$ is its position vector.

Significance of the Hamiltonian Operator

The Hamiltonian operator is central to quantum mechanics because it governs the time evolution of a quantum system through the time-dependent Schrödinger equation:

$$ i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r}, t) = \hat{H}\Psi(\mathbf{r}, t) $$

Where $\Psi(\mathbf{r}, t)$ is the wave function of the system.

Furthermore, the eigenvalues of the Hamiltonian operator represent the possible quantized energy levels of the system. When the Hamiltonian operator acts on a stationary state (an eigenfunction of $\hat{H}$), it simply returns the same function multiplied by a constant, which is the energy of that state. This is described by the time-independent Schrödinger equation:

$$ \hat{H}\psi(\mathbf{r}) = E\psi(\mathbf{r}) $$

This equation is fundamental for determining the allowed energy states of a quantum system.

Eigenvalues and Eigenfunctions

The concepts of eigenvalues and eigenfunctions are fundamental to understanding the results of measurements in quantum mechanics. When a quantum mechanical operator corresponding to an observable quantity acts on the wave function of a system, the result is often the same wave function multiplied by a scalar constant.

Definition

For a linear operator $\hat{A}$ and a wave function $\psi$, if the following relation holds:

$$ \hat{A}\psi = \lambda\psi $$

Then $\psi$ is called an **eigenfunction** (or eigenvector) of the operator $\hat{A}$, and the scalar constant $\lambda$ is called the corresponding **eigenvalue**.

The term "eigen" comes from German and means "own" or "characteristic." So, eigenfunctions are the "own functions" of an operator, and eigenvalues are their "own values."

In the context of quantum mechanics:

• If $\hat{A}$ is an operator for an observable quantity (e.g., energy, momentum, position), and $\psi$ is an eigenfunction of $\hat{A}$, then a measurement of that observable quantity on a system in the state described by $\psi$ will *always* yield the value $\lambda$.
• The eigenvalues $\lambda$ of Hermitian operators are always real numbers, which is consistent with the fact that physical observables must have real values.

Eigenvalues and Eigenfunctions of the Hamiltonian Operator

The most important application of eigenvalues and eigenfunctions in quantum mechanics is related to the Hamiltonian operator ($\hat{H}$). The equation:

$$ \hat{H}\psi = E\psi $$

Is known as the **time-independent Schrödinger equation**.

• The eigenfunctions ($\psi$) of the Hamiltonian operator are called **stationary states** or **energy eigenfunctions**. These represent states of the system with definite energy.
• The eigenvalues ($E$) of the Hamiltonian operator are the possible **energy levels** of the system. These are the only energy values that can be measured for the system.

For a system to be physically realistic, its wave functions must be well-behaved (e.g., continuous, single-valued, finite everywhere, and square-integrable). The solutions to the time-independent Schrödinger equation that satisfy these conditions are the allowed energy eigenfunctions and their corresponding energy eigenvalues.

Example: Particle in an Infinite Potential Well (1D)

Consider a particle of mass $m$ confined to a region $0 \le x \le L$, with an infinite potential energy outside this region. The potential $V(x)$ is:

$$ V(x) = \begin{cases} 0 & \text{for } 0 \le x \le L \\ \infty & \text{otherwise} \end{cases} $$

The Hamiltonian operator inside the well ($0 \le x \le L$) is $\hat{H} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}$, since $V(x)=0$. The time-independent Schrödinger equation becomes:

$$ -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi $$

This is a second-order ordinary differential equation. The general solution is of the form:

$$ \psi(x) = A \sin(kx) + B \cos(kx) $$

Where $k = \sqrt{\frac{2mE}{\hbar^2}}$.

Now, we apply boundary conditions. Since the potential is infinite outside the well, the wave function must be zero at the boundaries ($x=0$ and $x=L$) and everywhere outside.

1. At $x=0$: $\psi(0) = A \sin(0) + B \cos(0) = 0$. Since $\sin(0)=0$ and $\cos(0)=1$, this implies $B = 0$. So, $\psi(x) = A \sin(kx)$.
2. At $x=L$: $\psi(L) = A \sin(kL) = 0$. For a non-trivial solution ($A \neq 0$), we must have $\sin(kL) = 0$. This means $kL$ must be an integer multiple of $\pi$.

$$ kL = n\pi, \quad \text{where } n = 1, 2, 3, ... $$

Note that $n=0$ would lead to $k=0$ and $\psi(x)=0$, which is not a valid wave function (it cannot be normalized). Negative values of $n$ give the same physical states as positive values because $\sin(-n\pi x/L) = -\sin(n\pi x/L)$, and the negative sign can be absorbed into the amplitude $A$.

So, $k = \frac{n\pi}{L}$. Substituting this back into the definition of $k$:

$$ \frac{n\pi}{L} = \sqrt{\frac{2mE_n}{\hbar^2}} $$

Squaring both sides and solving for $E_n$:

$$ \left(\frac{n\pi}{L}\right)^2 = \frac{2mE_n}{\hbar^2} $$

$$ E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \quad \text{for } n = 1, 2, 3, ... $$

These are the quantized energy eigenvalues for the particle in a box.

The corresponding eigenfunctions are:

$$ \psi_n(x) = A \sin\left(\frac{n\pi x}{L}\right) $$

To find the normalization constant $A$, we use the condition that the total probability of finding the particle in the box is 1:

$$ \int_0^L |\psi_n(x)|^2 dx = 1 $$

$$ \int_0^L A^2 \sin^2\left(\frac{n\pi x}{L}\right) dx = 1 $$

Using the identity $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$:

$$ A^2 \int_0^L \frac{1}{2} \left(1 - \cos\left(\frac{2n\pi x}{L}\right)\right) dx = 1 $$

$$ \frac{A^2}{2} \left[ x - \frac{L}{2n\pi} \sin\left(\frac{2n\pi x}{L}\right) \right]_0^L = 1 $$

$$ \frac{A^2}{2} \left[ (L - 0) - (0 - 0) \right] = 1 $$

$$ \frac{A^2 L}{2} = 1 \implies A^2 = \frac{2}{L} \implies A = \sqrt{\frac{2}{L}} $$

So, the normalized energy eigenfunctions are:

$$ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right), \quad \text{for } n = 1, 2, 3, ... $$

Example: Eigenvalues of the Momentum Operator

Let's consider the momentum operator in one dimension, $\hat{p}_x = -i\hbar \frac{\partial}{\partial x}$. We want to find its eigenvalues and eigenfunctions. The equation is:

$$ \hat{p}_x \psi(x) = \lambda \psi(x) $$

$$ -i\hbar \frac{d\psi}{dx} = \lambda \psi $$

This is a first-order differential equation. Rearranging:

$$ \frac{d\psi}{\psi} = \frac{\lambda}{-i\hbar} dx = \frac{i\lambda}{\hbar} dx $$

Integrating both sides:

$$ \int \frac{d\psi}{\psi} = \int \frac{i\lambda}{\hbar} dx $$

$$ \ln(\psi) = \frac{i\lambda}{\hbar} x + C $$

$$ \psi(x) = e^{\frac{i\lambda}{\hbar} x + C} = e^C e^{\frac{i\lambda}{\hbar} x} $$

Let $A = e^C$. The eigenfunctions are of the form:

$$ \psi(x) = A e^{\frac{i\lambda}{\hbar} x} $$

The eigenvalues $\lambda$ are the possible values of momentum. In quantum mechanics, the momentum can take any real value, so $\lambda$ is a continuous variable. The eigenfunctions are complex exponential functions, representing plane waves.

A key issue with these eigenfunctions is that they are not normalizable over an infinite domain $(-\infty, \infty)$. In practice, one often uses "box normalization" or Dirac delta function normalization for such cases.

The eigenvalues of $\hat{p}_x$ are $\lambda$, which can be any real number. This means momentum is not quantized in the same way energy is for bound systems.