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Quantum Mechanics: Wave Functions
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Quantum Mechanics: Wave Functions

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Questions and Answers

What is the Schrödinger equation?

  • An equation that determines the probability of finding a particle at a coordinate
  • An equation that describes the dynamics of matter waves (correct)
  • An equation that calculates the energy levels of a particle in a box
  • An equation that describes the behavior of a free particle
  • What symbol is commonly used to represent the wave function in quantum mechanics?

    Ψ or ψ

    The presence of the imaginary number i means that the solutions to the Schrödinger equation are ______ quantities.

    complex

    The square of the absolute value of the wave function Ψ^2 tells us about the momentum of a particle.

    <p>False</p> Signup and view all the answers

    Match the following terms with their corresponding descriptions:

    <p>Potential well = A potential-energy function with a minimum Square-well potential = A simple model of an electron moving within a metallic sheet Potential barrier = A potential-energy function with a maximum Tunneling = When a particle encounters a barrier and may appear on the other side</p> Signup and view all the answers

    Study Notes

    Quantum Mechanics: Wave Functions

    • Quantum mechanics is the key to understanding the behavior of matter on the molecular, atomic, and nuclear scales.
    • It replaces the classical scheme of describing the state of a particle by its coordinates and velocity components.

    Wave Functions and the One-Dimensional Schrödinger Equation

    • The Schrödinger equation is the fundamental equation that describes the dynamics of matter waves.
    • A wave function is a mathematical description of a quantum state of a particle as a function of momentum, time, position, and spin, denoted by the Greek letter psi (Ψ or ψ).
    • Uppercase Ψ denotes a function of all space coordinates and time, while lowercase ψ denotes a function of space coordinates only, not time.

    Wave Functions: Particle Waves

    • Considering a free particle, the potential energy Ux has the same value for all x: U = 0.
    • The wave function for a free particle is described by the angular frequency and wave number.

    One-Dimensional Schrödinger Equation

    • Developed in 1926 by Erwin Schrödinger, the equation is used to find the quantum mechanical wave function that satisfies it for a particular situation.
    • The presence of the imaginary number i means that the solutions to the Schrödinger equation are complex quantities, with a real part and an imaginary part.

    Interpreting Wave Functions

    • The complex nature of the wave function for a free particle makes it challenging to interpret.
    • The square of the absolute value of the wave function (Ψ²) of a particle at each point tells us about the probability of finding the particle around that point.
    • Ψ(x, t)² dx is the probability that the particle will be found at time t at a coordinate in the range from x to x + dx, first made by Max Born.

    One-Dimensional Schrödinger Equation with Potential Energy

    Time-Independent One-Dimensional Schrödinger Equation

    Particle in a Box

    • The idea of the particle in a box is to determine, for a given potential energy function U(x), the possible stationary state wave functions, and the corresponding energies E.
    • This model might represent an electron that is free to move within a long, straight molecule or along a very thin wire.

    Energy Levels for a Particle in a Box

    • The energy levels for a particle in a box can be found by solving the Schrödinger equation.

    Wave Function for a Particle in a Box

    • The wave function for a particle in a box is a solution to the Schrödinger equation.

    Potential Wells and Barriers

    • A potential well is a potential-energy function U(x) that has a minimum.
    • A potential barrier is the opposite of a potential well, with a maximum potential energy.
    • Tunneling occurs when a quantum-mechanical particle encounters a barrier, even if it has less energy, it may appear on the other side.

    Tunneling

    • The tunneling probability T that the particle gets through the barrier is proportional to the square of the ratio of the amplitudes of the sinusoidal wave functions on the two sides of the barrier.
    • Applications of tunneling include tunnel diodes, Josephson junctions, scanning tunneling microscopes (STM), electron tunneling in enzymes, and fusion reactions.

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    Related Documents

    5. Quantum Mechanics.pdf

    Description

    Understand the behavior of matter at molecular, atomic, and nuclear scales through quantum mechanics, replacing classical descriptions with wave functions and the Schrödinger equation.

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