The Simple Harmonic Oscillator

Oct 13, 2025 by pynkocean.com 68 views

Introduction

In the realm of quantum mechanics, the simple harmonic oscillator is a fundamental system that has been extensively studied. It is a crucial model for understanding the behavior of particles in a potential that is proportional to the square of their displacement from a reference point. The harmonic oscillator is a closed system, meaning that its energy remains constant over time. However, when an external potential is applied to the system, its energy can change. In this article, we will explore how a finite time external potential can alter the energy of a simple harmonic oscillator.

The Simple Harmonic Oscillator

The simple harmonic oscillator is a quantum mechanical system that consists of a particle of mass m, subject to a potential energy given by:

U(x) = (1/2)kx^2

where k is the spring constant and x is the displacement from the equilibrium position. The Hamiltonian of the system is given by:

H = (p^2)/(2m) + (1/2)kx^2

where p is the momentum of the particle. The energy eigenstates of the harmonic oscillator are given by:

E_n = (n + 1/2)ħω

where n is a non-negative integer, ħ is the reduced Planck constant, and ω is the angular frequency of the oscillator.

External Potential

When an external potential is applied to the harmonic oscillator, its energy can change. The external potential can be time-dependent or time-independent. In this article, we will consider a time-independent external potential. The total Hamiltonian of the system is given by:

H = (p^2)/(2m) + (1/2)kx^2 + V(x)

where V(x) is the external potential.

Finite Time External Potential

A finite time external potential is a potential that is applied to the system for a finite period of time. The external potential can be turned on and off at specific times. The total Hamiltonian of the system is given by:

H = (p^2)/(2m) + (1/2)kx^2 + V(x) * θ(t - t_0) * θ(t_1 - t)

where θ(t) is the Heaviside step function, t_0 is the time when the external potential is turned on, and t_1 is the time when the external potential is turned off.

Effect of Finite Time External Potential on Energy

The effect of a finite time external potential on the energy of the harmonic oscillator can be studied using the time-dependent Schrödinger equation. The time-dependent wave function of the system is given by:

ψ(x, t) = ∑_n c_n ψ_n(x) e^(-iE_nt/ħ)

where ψ_n(x) is the energy eigenstate of the harmonic oscillator, E_n is the energy eigenvalue, and c_n is the coefficient of the energy eigenstate.

When the external potential is applied to the system, the energy eigenstates of the harmonic oscillator are no longer eigenstates of the total Hamiltonian. The energy eigenvalues of the system are given by:

E_n = (n + 1/2)ħω + ∫_0^t V(x) * ψ_n(x) dx

where the integral is evaluated over the time period when the external potential is applied.

Numerical Results

To study the effect of a finite time external potential on the energy of the harmonic oscillator, we can use numerical methods to solve the time-dependent Schrödinger equation. The numerical results can be used to study the behavior of the system over time.

Conclusion

In conclusion, a finite time external potential can alter the energy of a simple harmonic oscillator. The energy eigenstates of the harmonic oscillator are no longer eigenstates of the total Hamiltonian when the external potential is applied. The energy eigenvalues of the system are given by the sum of the energy eigenvalue of the harmonic oscillator and the integral of the external potential over the time period when it is applied. The numerical results can be used to study the behavior of the system over time.

References

[1] Landau, L. D., & Lifshitz, E. M. (1977). Quantum mechanics: Non-relativistic theory. Pergamon Press.
[2] Messiah, A. (1961). Quantum mechanics. North-Holland Publishing Company.
[3] Dirac, P. A. M. (1958). The principles of quantum mechanics. Oxford University Press.

Appendix

The time-dependent Schrödinger equation is given by:

iħ(∂ψ/∂t) = Hψ

where ψ is the time-dependent wave function of the system, H is the total Hamiltonian of the system, and i is the imaginary unit.

The time-dependent Schrödinger equation can be solved numerically using various methods, such as the finite difference method or the finite element method. The numerical results can be used to study the behavior of the system over time.

Mathematical Derivations

The energy eigenstates of the harmonic oscillator are given by:

ψ_n(x) = (1/√2^n n!) (a^†)n ψ_0(x)

where a^† is the creation operator, ψ_0(x) is the ground state wave function of the harmonic oscillator, and n is a non-negative integer.

The energy eigenvalues of the harmonic oscillator are given by:

E_n = (n + 1/2)ħω

The time-dependent wave function of the system is given by:

ψ(x, t) = ∑_n c_n ψ_n(x) e^(-iE_nt/ħ)

where c_n is the coefficient of the energy eigenstate.

E_n = (n + 1/2)ħω + ∫_0^t V(x) * ψ_n(x) dx

where the integral is evaluated over the time period when the external potential is applied.

Code

The numerical results can be obtained using various programming languages, such as Python or MATLAB. The code can be used to solve the time-dependent Schrödinger equation numerically and study the behavior of the system over time.

import numpy as np
from scipy.integrate import quad
def harmonic_oscillator_wave_function(x, n):
return (1/np.sqrt(2**n * np.math.factorial(n))) * (np.sqrt(m) * x)n * np.exp(-m * x2 / (2 * hbar))
def external_potential(x, t):
return V * np.exp(-((x - x_0)2) / (2 * sigma2))
def time_dependent_wave_function(x, t, n):
return harmonic_oscillator_wave_function(x, n) * np.exp(-1j * E_n * t / hbar)
def total_energy(n, t):
return E_n + quad(lambda x: external_potential(x, t) * harmonic_oscillator_wave_function(x, n), 0, np.inf)[0]
m = 1  # mass
k = 1  # spring constant
hbar = 1  # reduced Planck constant
V = 1  # external potential strength
x_0 = 1  # external potential center
sigma = 1  # external potential width
E_n = (n + 1/2) * hbar  # energy eigenvalue
t = 1  # time
total_energy_value = total_energy(n, t)
print("Total energy:", total_energy_value)

Q: What is a simple harmonic oscillator?

A: A simple harmonic oscillator is a quantum mechanical system that consists of a particle of mass m, subject to a potential energy given by U(x) = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position.

Q: What is the Hamiltonian of a simple harmonic oscillator?

A: The Hamiltonian of a simple harmonic oscillator is given by H = (p^2)/(2m) + (1/2)kx^2, where p is the momentum of the particle.

Q: What is the energy eigenstate of a simple harmonic oscillator?

A: The energy eigenstate of a simple harmonic oscillator is given by ψ_n(x) = (1/√2^n n!) (a^†)n ψ_0(x), where a^† is the creation operator, ψ_0(x) is the ground state wave function of the harmonic oscillator, and n is a non-negative integer.

Q: What is the energy eigenvalue of a simple harmonic oscillator?

A: The energy eigenvalue of a simple harmonic oscillator is given by E_n = (n + 1/2)ħω, where ħ is the reduced Planck constant and ω is the angular frequency of the oscillator.

Q: What is a finite time external potential?

A: A finite time external potential is a potential that is applied to the system for a finite period of time. The external potential can be turned on and off at specific times.

Q: How does a finite time external potential affect the energy of a simple harmonic oscillator?

A: A finite time external potential can alter the energy of a simple harmonic oscillator. The energy eigenstates of the harmonic oscillator are no longer eigenstates of the total Hamiltonian when the external potential is applied. The energy eigenvalues of the system are given by E_n = (n + 1/2)ħω + ∫_0^t V(x) * ψ_n(x) dx, where the integral is evaluated over the time period when the external potential is applied.

Q: How can the effect of a finite time external potential on the energy of a simple harmonic oscillator be studied?

A: The effect of a finite time external potential on the energy of a simple harmonic oscillator can be studied using the time-dependent Schrödinger equation. The time-dependent wave function of the system is given by ψ(x, t) = ∑_n c_n ψ_n(x) e^(-iE_nt/ħ), where c_n is the coefficient of the energy eigenstate.

Q: What is the total Hamiltonian of a simple harmonic oscillator with a finite time external potential?

A: The total Hamiltonian of a simple harmonic oscillator with a finite time external potential is given by H = (p^2)/(2m) + (1/2)kx^2 + V(x) * θ(t - t_0) * θ(t_1 - t), where θ(t) is the Heaviside step function, t_0 is the time when the external potential is turned on, and t_1 is the time when the external potential is turned off.

Q: How can the numerical results be obtained?

A: The numerical results can be obtained using various programming languages, such as Python or MATLAB. The code can be used to solve the time-dependent Schrödinger equation numerically and study the behavior of the system over time.

Q: What are the limitations of the current study?

A: The current study assumes that the external potential is time-independent and that the system is in a non-degenerate energy eigenstate. The study also assumes that the system is in a closed system, meaning that the energy remains constant over time. In reality, the system may be in a mixed state or in an open system, which can affect the results.

Q: What are the future directions of the study?

A: The future directions of the study include investigating the effect of a finite time external potential on the energy of a simple harmonic oscillator in a mixed state or in an open system. The study also includes investigating the effect of a finite time external potential on the energy of a simple harmonic oscillator with a time-dependent external potential.

Q: What are the applications of the study?

A: The study has applications in various fields, including quantum mechanics, condensed matter physics, and materials science. The study can be used to understand the behavior of particles in a potential that is proportional to the square of their displacement from a reference point. The study can also be used to design and optimize materials with specific properties.

Q: What are the implications of the study?

A: The study has implications for our understanding of the behavior of particles in a potential that is proportional to the square of their displacement from a reference point. The study also has implications for the design and optimization of materials with specific properties. The study can be used to develop new materials and technologies with improved performance and efficiency.