Schrödinger Wave Equation for a Free Particle

Basic Wave Equation and Definitions

The motion of a free particle in quantum mechanics is described by a wave function, which encapsulates the probability amplitude of finding the particle at a given position and time. The momentum and energy of the particle are related to the wave properties.

• Wave Vector (k): , where is the wavelength.

• Propagation Constant: is also known as the propagation constant.

• Energy Expression: , where is the frequency.

• Plane Monochromatic Wave: The simplest wave is described by .

Example: A free particle moving along the x-axis with velocity has a wave function .

Time-Dependent Schrödinger Equation

Derivation and Application

The time-dependent Schrödinger equation describes the evolution of the wave function for a particle with kinetic and potential energy.

• General Form:

• Hamiltonian Operator:

• Potential Energy Function: can be time-dependent or independent.

Example: For a free particle (), the equation simplifies to .

Time-Independent Schrödinger Equation

Separation of Variables and Stationary States

When the potential is independent of time, the Schrödinger equation can be solved by separation of variables.

• Product Solution:

• Time-Independent Equation:

• Stationary State: A solution where is independent of time.

Example: The stationary state is defined as .

Operators, Eigenfunctions, and Observables

Definitions and Properties

Operators in quantum mechanics act on wave functions to yield physical observables.

• Eigenvalue Equation:

• Observables: Quantities measured in experiments, such as position and momentum.

• Commutator:

Example: The position and momentum operators do not commute: .

Orthogonality and Normalization of Eigenfunctions

Mathematical Structure

Eigenfunctions corresponding to different eigenvalues are orthogonal and can be normalized.

• Orthogonality Condition: for

• Normalization Condition:

Example: Normalized eigenfunctions are called orthonormal.

Expectation Values

Calculation and Interpretation

The expectation value of an observable is the average value obtained from many measurements.

• General Formula:

• Position:

• Momentum:

Example: For a normalized wave function, gives the average position.

Energy, Momentum, and Other Operators

Operator Forms

Operators are used to represent physical quantities in quantum mechanics.

• Energy Operator:

• Momentum Operator:

• Kinetic Energy Operator:

• Potential Energy Operator:

• Position Operator:

Example: The velocity operator is .

Application: Particle in a 1D Box

Schrödinger Equation and Quantization

The particle in a box is a fundamental quantum system with quantized energy levels.

• Potential: for , otherwise.

• Wave Function:

• Energy Levels:

• Normalization:

Example: The probability density is .

Commutator Relations and Methods

Commutator Calculations

Commutators are used to determine the relationships between operators.

• Position and Momentum:

• Other Relations: ,

Example: Commutator calculations for more complex operators.

Numerical Examples

Expectation Values for Specific Wave Functions

Calculations of expectation values for given wave functions illustrate the application of operators.

• Wave Function:

• Expectation Value of Momentum:

• Expectation Value of :

Example: Calculation for yields , .

Summary Table: Common Quantum Operators

Operator and Observable Correspondence

Additional info: These notes cover the foundational aspects of quantum mechanics relevant to college-level physics, including the Schrödinger equation, operators, eigenfunctions, commutators, and applications such as the particle in a box.