⚛️Intro to Quantum Mechanics I Unit 8 – Quantum Harmonic Oscillator

Key Concepts

• Quantum harmonic oscillator (QHO) is a fundamental model in quantum mechanics that describes a particle confined by a parabolic potential
• Exhibits discrete energy levels, unlike the continuous energy spectrum of a classical harmonic oscillator
• Ground state energy is non-zero, given by $E_0 = \frac{1}{2}\hbar\omega$, where $\hbar$ is the reduced Planck's constant and $\omega$ is the angular frequency
• Energy levels are evenly spaced, with a separation of $\hbar\omega$ between consecutive levels
• Wavefunctions for each energy level are described by Hermite polynomials multiplied by a Gaussian function
• Position and momentum operators do not commute, leading to the Heisenberg uncertainty principle
• Expectation values of position and momentum are zero for all energy levels, but their variances are non-zero and depend on the energy level

Classical vs Quantum Oscillators

• Classical harmonic oscillator has a continuous energy spectrum, while the quantum harmonic oscillator has discrete energy levels
• In the classical case, the particle can have any energy value, whereas in the quantum case, the particle can only occupy specific energy levels
• Classical oscillator's energy can be zero, but the quantum oscillator has a non-zero ground state energy
• Classical oscillator's position and momentum can be simultaneously determined with arbitrary precision, while the quantum oscillator is subject to the Heisenberg uncertainty principle
• As the quantum number increases, the behavior of the quantum oscillator approaches that of the classical oscillator (correspondence principle)
• Classical oscillator's probability distribution is concentrated at the turning points, while the quantum oscillator's probability distribution is more evenly spread out

Mathematical Framework

• Schrödinger equation for the quantum harmonic oscillator: $-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}m\omega^2x^2\psi = E\psi$
  • $m$ is the mass of the particle, $\omega$ is the angular frequency, and $x$ is the position
• Energy eigenvalues: $E_n = \left(n + \frac{1}{2}\right)\hbar\omega$, where $n = 0, 1, 2, ...$
• Wavefunctions: $\psi_n(x) = \frac{1}{\sqrt{2^n n!}}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-\frac{m\omega x^2}{2\hbar}} H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right)$
  • $H_n(x)$ are the Hermite polynomials
• Creation and annihilation operators: $\hat{a}^{\dagger} = \sqrt{\frac{m\omega}{2\hbar}}\left(\hat{x} - \frac{i}{m\omega}\hat{p}\right)$ and $\hat{a} = \sqrt{\frac{m\omega}{2\hbar}}\left(\hat{x} + \frac{i}{m\omega}\hat{p}\right)$
  • These operators raise or lower the energy level by one quantum

Energy Levels and Wavefunctions

• Ground state (n = 0) has the lowest energy, $E_0 = \frac{1}{2}\hbar\omega$, and a Gaussian wavefunction, $\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-\frac{m\omega x^2}{2\hbar}}$
• Excited states (n > 0) have energies $E_n = \left(n + \frac{1}{2}\right)\hbar\omega$ and wavefunctions $\psi_n(x)$ that are Hermite polynomials multiplied by the Gaussian function
• Wavefunctions are orthonormal, meaning $\int_{-\infty}^{\infty} \psi_n^*(x) \psi_m(x) dx = \delta_{nm}$, where $\delta_{nm}$ is the Kronecker delta
• Probability density of finding the particle at position $x$ in the $n$-th energy level is given by $|\psi_n(x)|^2$
• As the energy level increases, the number of nodes in the wavefunction also increases
• Expectation value of the potential energy is equal to the expectation value of the kinetic energy for all energy levels

Operators and Observables

• Position operator: $\hat{x} = x$
• Momentum operator: $\hat{p} = -i\hbar\frac{d}{dx}$
• Hamiltonian operator: $\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}m\omega^2x^2$
• Commutation relation between position and momentum operators: $[\hat{x}, \hat{p}] = i\hbar$
  • This leads to the Heisenberg uncertainty principle, $\Delta x \Delta p \geq \frac{\hbar}{2}$
• Expectation values of position and momentum are zero for all energy levels: $\langle \hat{x} \rangle = \langle \hat{p} \rangle = 0$
• Variances of position and momentum depend on the energy level: $\Delta x = \sqrt{\frac{\hbar}{2m\omega}(2n+1)}$ and $\Delta p = \sqrt{\frac{m\hbar\omega}{2}(2n+1)}$

Applications and Examples

• Vibrations of diatomic molecules can be modeled as quantum harmonic oscillators
  • The energy levels correspond to different vibrational states of the molecule
• Phonons in solid-state physics are quantized lattice vibrations that can be described by the quantum harmonic oscillator model
• Electromagnetic field in a cavity can be treated as a collection of quantum harmonic oscillators
  • Each mode of the field corresponds to a different oscillator with a specific frequency
• Quantum harmonic oscillator is a useful approximation for many potentials near their minimum, such as the Morse potential for molecular vibrations
• Quantum dots, nanoscale semiconductor structures, can be modeled as two-dimensional or three-dimensional quantum harmonic oscillators
• Quantum harmonic oscillator is a key component in the quantum theory of radiation, describing the interaction between light and matter

Problem-Solving Techniques

• Identify the given information, such as mass, angular frequency, and quantum number, and the desired quantity to be calculated
• Use the appropriate formula for the energy eigenvalues, wavefunctions, or expectation values based on the given information
• Substitute the given values into the formula and perform the necessary calculations
• For problems involving the Schrödinger equation, use the differential equation and boundary conditions to solve for the energy eigenvalues and wavefunctions
• When dealing with operators, use the commutation relations and eigenvalue equations to simplify the expressions
• Apply the Heisenberg uncertainty principle when discussing the limitations on simultaneously measuring position and momentum
• Use the properties of Hermite polynomials and Gaussian functions when manipulating wavefunctions

Connections to Advanced Topics

• Quantum harmonic oscillator is a fundamental building block for many advanced quantum systems, such as coupled oscillators and quantum fields
• Creation and annihilation operators introduced in the quantum harmonic oscillator formalism are essential in the second quantization formalism used in quantum field theory
• Coherent states, which are quantum states that minimize the Heisenberg uncertainty principle, are closely related to the quantum harmonic oscillator
• Squeezed states, another type of quantum state with reduced uncertainty in one observable at the expense of increased uncertainty in the conjugate observable, can be generated using the quantum harmonic oscillator formalism
• The quantum harmonic oscillator is a useful tool for understanding the concept of entanglement in quantum mechanics
• Generalizations of the quantum harmonic oscillator, such as the anharmonic oscillator and the Morse potential, are used to model more complex systems in quantum chemistry and condensed matter physics
• The quantum harmonic oscillator is a key example in the study of supersymmetric quantum mechanics, where partner potentials are related by supersymmetry transformations