4.1 Schrödinger Equation and Wave Functions

Quantum mechanics is all about understanding tiny particles. The Schrödinger equation is the key to this world, describing how these particles behave. It's like a magic formula that unlocks the secrets of atoms and molecules.

Wave functions are the heart of quantum mechanics. They're mathematical descriptions of particles that tell us where they might be and how they move. By solving the Schrödinger equation, we can find these wave functions and learn about energy levels in atoms.

Schrödinger Equation and Its Meaning

Physical Interpretation

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• The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum-mechanical system (electron in an atom or molecule)
• The time-dependent Schrödinger equation describes the evolution of a quantum system over time
• The time-independent Schrödinger equation describes the stationary states of a quantum system

Components of the Schrödinger Equation

• The Hamiltonian operator (Ĥ) represents the total energy of the system
  • Includes both kinetic and potential energy terms
• The wave function (Ψ) is a mathematical function that contains all the information about the quantum state of a system
  • Complex-valued function of position and time
• The eigenvalues of the Hamiltonian operator correspond to the allowed energy levels of the quantum system
• The eigenfunctions represent the corresponding stationary states or orbitals

Solving the Time-Independent Schrödinger Equation

Simple Quantum Systems

• The time-independent Schrödinger equation is an eigenvalue equation of the form ĤΨ = EΨ
  • E is the energy eigenvalue
  • Ψ is the corresponding eigenfunction or stationary state
• The particle in a box model is a simple quantum system
  • Particle is confined to a one-dimensional box with infinite potential walls
  • Solutions are standing waves with quantized energy levels
• The quantum harmonic oscillator models the behavior of a particle in a parabolic potential well
  • Solutions involve Hermite polynomials and quantized energy levels

Hydrogen Atom

• The hydrogen atom is a fundamental quantum system consisting of a proton and an electron
• Solutions to the Schrödinger equation for the hydrogen atom involve:
  • Spherical harmonics
  • Associated Laguerre polynomials
• Solutions lead to the atomic orbitals and energy levels

Properties of Wave Functions

Physically Acceptable Solutions

• Wave functions must be continuous, single-valued, and square-integrable to represent physically acceptable solutions to the Schrödinger equation

Normalization

• Normalization is the process of scaling a wave function
  • Probability of finding the particle somewhere in space is equal to 1
• The normalization condition is expressed as the integral of the squared modulus of the wave function over all space being equal to 1

Orthogonality

• Orthogonality is a property of wave functions corresponding to different quantum states
• Two wave functions are orthogonal if their overlap integral is equal to zero
  • Overlap integral is the integral of the product of the two functions over all space
• The Kronecker delta function expresses the orthogonality of wave functions
  • Equal to 1 if the two quantum states are the same
  • Equal to 0 if they are different

Born Interpretation for Probabilities

Probability Density

• The Born interpretation states that the probability of finding a particle in a given region of space is proportional to the squared modulus of the wave function integrated over that region
• The probability density is defined as the squared modulus of the wave function, |Ψ(x,t)|^2
  • Represents the probability of finding the particle at a specific position x at time t

Calculating Probabilities

• To calculate the probability of finding a particle in a given region, integrate the probability density over that region
  • Probability is given by the integral of |Ψ(x,t)|^2 dx over the specified region
• The expectation value of an observable (position, momentum, energy) can be calculated using the wave function and the corresponding operator
  • Expectation value is given by the integral of Ψ* Ô Ψ dx, where Ô is the operator associated with the observable