What is a quantum harmonic oscillator?

Question

Olivia Askew · Accepted Answer

The quantum harmonic oscillator is essentially a two-body problem consisting of two solid spheres connected by a spring.

A basic version of it is called the simple harmonic oscillator (SHO), in which the spring has no dampening factor (no anharmonicity constant):

Since there is no dampening factor, the energy levels are evenly spaced, separated by #ℏomega#, or #hnu# (the anharmonic oscillator would have quadratic convergence of the energy levels). Furthermore, the lowest energy level is #1/2ℏomega#, and not #0#.

The Hamiltonian operator for the SHO system in one dimension is:

#color(blue)(hatH_"SHO") = hatK + hatV#
#= -ℏ^2/(2mu)d^2/(dx^2) + 1/2kx^2#
#= [-iℏd/(dx)]^2/(2mu) + 1/2kx^2#
#= color(blue)(hatp^2/(2mu) + 1/2kx^2)#
where #mu = (m_1m_2)/(m_1 + m_2)# is the reduced mass, #hatp# is the momentum operator, #k# is the force constant, and #x# is the relative displacement from equilibrium. #hatK# and #hatV# were the kinetic and potential energy operators.
The normalized wave function for the #upsilon#th energy level in general is the product of a Hermite polynomial and a decaying exponential.

#color(blue)(psi_(upsilon)(x) = N_(upsilon)H_(upsilon)(sqrtalphax)e^(-alphax^2"/"2))#
where:

#N_(upsilon) = [1/(2^(upsilon) upsilon!)(alpha/(pi))^"1/2"]^"1/2"# is the normalization constant.

#H_(upsilon)(sqrtalphax) = (-1)^(upsilon)e^(-alphax^2)d^(upsilon)/(d (sqrtalphax)^(upsilon))[e^(-alphax^2)]# is the Hermite polynomial.

#alpha = sqrt((kmu)/(ℏ^2))# is a variable defined for convenience of expressing the function.

Applying the variational method on some trial wave function, #psi_0(x) = Ne^(-cx^2)#, where #N = ((2c)/pi)^"1/4"# upon normalization, would give you:

#color(blue)(E_0 = 1/2ℏomega)#

which is in general,

#bb(E_(upsilon) = ℏomega(upsilon + 1/2))#,
as shown in the first image.

Olivia Askew · Answer

undefined A quantum harmonic oscillator is a quantum mechanical system that describes a particle (or field) in a potential well with a quadratic potential energy function. It's a fundamental model used in quantum mechanics to study vibrations, atomic and molecular bonds, and various other physical systems. It's characterized by discrete energy levels, with each energy level separated by a constant amount known as a quantum of energy. This system is essential for understanding many aspects of quantum theory, such as the behavior of elementary particles and the properties of solids.

Nora Arzate · Answer

undefined A quantum harmonic oscillator is a theoretical model used in quantum mechanics to describe the behavior of certain physical systems, such as atomic vibrations and molecular vibrations. It consists of a particle (or system) that experiences a restoring force proportional to its displacement from a fixed equilibrium position. This force is typically represented by a quadratic potential energy function. In the context of quantum mechanics, the behavior of the harmonic oscillator is described using the Schrödinger equation.

The solutions to the Schrödinger equation for the quantum harmonic oscillator yield a set of quantized energy levels, rather than a continuous spectrum. These energy levels are evenly spaced and non-degenerate, meaning that each energy level corresponds to a unique quantum state. The energy of the quantum harmonic oscillator is thus said to be "quantized."

The lowest energy level, known as the ground state, has a non-zero energy due to the zero-point energy, a consequence of Heisenberg's uncertainty principle. As the energy increases, the quantum harmonic oscillator transitions through higher energy levels, with each level separated by a fixed amount known as a quantum of energy.

The solutions to the Schrödinger equation for the quantum harmonic oscillator also yield wave functions, which describe the probability amplitude of finding the particle at a given position. These wave functions exhibit characteristic patterns of oscillation, reflecting the harmonic nature of the potential energy function.

The quantum harmonic oscillator has wide-ranging applications in various fields of physics, including quantum chemistry, solid-state physics, and particle physics. It provides a fundamental framework for understanding the behavior of vibrational modes in molecules, phonons in solids, and certain atomic and subatomic systems. Additionally, the quantum harmonic oscillator serves as a cornerstone in the development of quantum mechanics, offering insights into the principles of quantization and wave-particle duality.