8.2 Quantum harmonic oscillator Hamiltonian and energy levels

The quantum harmonic oscillator is a crucial model in quantum mechanics. It describes a particle in a parabolic potential well, like a mass on a spring at the atomic level. This system reveals key quantum features like energy quantization and zero-point energy.

The Hamiltonian for this system combines kinetic and potential energy terms. Solving the Schrödinger equation with this Hamiltonian yields discrete energy levels and wave functions. These solutions use Hermite polynomials and show how energy is quantized in quantum systems.

Hamiltonian and Schrödinger Equation

Quantum Harmonic Oscillator Fundamentals

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• Hamiltonian operator represents total energy of quantum harmonic oscillator system combines kinetic and potential energy terms
• Hamiltonian for quantum harmonic oscillator expressed as $H = \frac{p^2}{2m} + \frac{1}{2}kx^2$

• p

  denotes momentum operator,

  m

  represents mass of oscillating particle,

  k

  signifies spring constant, and

  x

  indicates position operator
• Schrödinger equation for harmonic oscillator written as $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}kx^2\psi = E\psi$

• ψ

  represents wave function,

  E

  denotes energy eigenvalue, and

  ℏ

  stands for reduced Planck's constant
• Solutions to Schrödinger equation yield wave functions describing quantum states of harmonic oscillator

Mathematical Tools and Solutions

• Hermite polynomials play crucial role in solving Schrödinger equation for quantum harmonic oscillator
• Hermite polynomials defined by recursive relation $H_{n+1}(x) = 2xH_n(x) - 2nH_{n-1}(x)$
• First few Hermite polynomials include $H_0(x) = 1$, $H_1(x) = 2x$, $H_2(x) = 4x^2 - 2$
• Wave functions for quantum harmonic oscillator expressed using Hermite polynomials and Gaussian function
• General form of wave function given by $\psi_n(x) = N_n H_n(\alpha x) e^{-\alpha^2 x^2/2}$

• Nn

  represents normalization constant,

  α

  depends on mass and angular frequency of oscillator

Energy Levels and Eigenvalues

Quantization of Energy in Harmonic Oscillator

• Quantized energy levels result from discrete solutions to Schrödinger equation for quantum harmonic oscillator
• Energy eigenvalues for quantum harmonic oscillator given by $E_n = \hbar \omega (n + \frac{1}{2})$

• ω

  represents angular frequency of classical oscillator,

  n

  denotes quantum number (non-negative integer)
• Quantum number

  n

  determines energy state of oscillator ranges from 0 to infinity
• Energy levels equally spaced with separation of

  ℏω

  between adjacent levels
• Quantum harmonic oscillator exhibits discrete energy spectrum unlike classical counterpart with continuous energy

Fundamental Energy Concepts

• Zero-point energy refers to lowest possible energy state of quantum harmonic oscillator
• Zero-point energy given by $E_0 = \frac{1}{2}\hbar \omega$ corresponds to ground state (n = 0)
• Zero-point energy arises from Heisenberg uncertainty principle prevents particle from being completely at rest
• Energy eigenvalues increase linearly with quantum number

  n

• Probability distribution of particle's position in each energy state described by square of wave function
• Higher energy states exhibit more nodes in wave function correspond to more complex oscillation patterns