7- CHEM-327 Lecture 7: Quantum Mechanics - Wave Functions, Operators, and Eigenvalues

Time-Independent Wave Function for a Free Electron

  • A free electron is defined by having no potential energy, meaning $V(x,t) = 0$.

  • Under this condition, a solution to the time-independent Schrödinger equation exists, describing the behavior of such an electron.

Normalization of Wave Functions

  • Physical Interpretation: The probability of finding a particle "somewhere" in space must be equal to one.

  • Mathematical Condition: This is expressed as the integral of the square of the magnitude of the wave function over all space:
    ∫−∞∞∣Ψ(x,t)∣2dx=1\int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx = 1

  • Limiting Cases: Many theoretical wave functions, particularly those describing free particles (like the free electron wave function), are limiting cases that do not inherently fulfill this normalization condition over infinite space.

  • Practical Approach: To normalize such wave functions, or any wave function that doesn't inherently satisfy the condition, one must consider it non-zero only for a specific, finite range of x values, or multiply it by a normalization constant.

  • Normalization Process: If the integral of the squared magnitude of an unnormalized wave function $\Psi{unnormalized}(x)$ is equal to a constant $N$ (i.e., $\int{-\infty}^{\infty} |\Psi{unnormalized}(x)|^2 dx = N$), then the normalized wave function $\Psi{normalized}(x)$ is given by:
    Ψ<em>normalized(x)=1NΨ</em>unnormalized(x)\Psi<em>{normalized}(x) = \frac{1}{\sqrt{N}} \Psi</em>{unnormalized}(x)

Quantum Mechanical Operators

  • The Schrödinger equation can be expressed in a concise shorthand notation:
    H^Ψ=EΨ\hat{H}\Psi = E\Psi

  • Definition: An operator is a mathematical instruction or rule that is applied to a wave function. It transforms one function into another.

  • Hamiltonian Operator ($\hat{H}$): In the context of the Schrödinger equation, $\hat{H}$ is specifically referred to as the Hamiltonian operator.

  • Application: Operators are meant to be applied to, or operate on, a wave function, for example, $\hat{A}\Psi(x)$.

Eigenvalues and Eigenfunctions

  • General Equation: An equation of the form $\hat{A}\Psi = c\Psi$ is fundamental in quantum mechanics.

  • Operator ($\hat{A}$): This is the quantum mechanical operator being applied.

  • Eigenvalue (c): This is a constant numerical value obtained from the application of the operator. It is a concept rooted in linear algebra, though a deep understanding of linear algebra is not required to grasp its significance here.

  • Eigenfunction ($\Psi$): Any wave function that satisfies the eigenvalue equation (i.e., when an operator acts on it, it returns the same function multiplied by a constant) is called an eigenfunction of that operator.

  • Example: To find an eigenfunction of the operator $\frac{d}{dx}$ with an eigenvalue of $2$:

    • We set up the eigenvalue equation: $\frac{d}{dx}\Psi(x) = 2\Psi(x)$

    • The solution to this differential equation is of the form: $\Psi(x) = Ce^{2x}$, where $C$ is an arbitrary constant.

The Interpretation of Operators and Eigenvalues

  • Operators and Measurable Quantities: A fundamental principle of quantum mechanics is that every measurable physical quantity (e.g., energy, momentum, position) corresponds to a unique quantum mechanical operator.

  • Eigenvalues and Measured Values: The eigenvalues obtained from applying an operator to a wave function represent the possible measured values for the physical quantity associated with that operator. These are the sharp, definite values that would be experimentally observed.

  • Hamiltonian Operator and Energy: The Hamiltonian operator ($\hat{H}$) from the Schrödinger equation corresponds specifically to the total energy of the system. Its eigenvalue for a given wave function represents the energy of the particle described by that wave function.

Kinetic and Potential Energy Operators

  • The Hamiltonian operator ($\hat{H}$), which represents the total energy, is conventionally composed of two main parts:

    • The kinetic energy operator ($\hat{T}$)

    • The potential energy operator ($\hat{V}$)

  • Thus, the Hamiltonian can be written as: $\hat{H} = \hat{T} + \hat{V}$

  • Kinetic Energy Operator (1D): In one dimension, the kinetic energy operator is given by:
    T^=−ℏ22m∂2∂x2\hat{T} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}
    where $\hbar$ is the reduced Planck constant and $m$ is the mass of the particle.

  • Potential Energy Operator (1D): The potential energy operator is simply multiplication by the potential energy function:
    V^=V(x)\hat{V} = V(x)

  • Example: To find the kinetic energy for the free electron wave function, one would apply the kinetic energy operator $\hat{T}$ to the specific free electron wave function $\Psi(x)$. The resulting eigenvalue would be the kinetic energy.

Normalization Question Example (Conceptual)

  • Problem: Given an unnormalized wave function $\Psi(x)$ and the result of the integral $\int_{-\infty}^{\infty} |\Psi(x)|^2 dx$. The task is to find the normalized wave function.

  • Solution Principle: To normalize a wave function, you take the square root of the integral of the squared magnitude of $\Psi(x)$, and then divide the original $\Psi(x)$ by this value (which is the normalization constant). If the integral evaluates to a value $N$, the normalized wave function will be $\frac{1}{\sqrt{N}}\Psi(x)$.