Rating: **4.2**/5.0 (26 Votes)

Category: Homework

The harmonic oscillator is where the force is proportional to the displacement. A physical implementation of the one-dimensional h.o. is the spring pendulum. The normal pendulum isn't a harmonic oscillator (but can be approximated as one if the amplitude is sufficiently low). Solving a spherical pendulum is *much* harder than solving a harmonic oscillator, so you'd not normally substitute a spherical pendulum for a harmonic oscillator (rather, the other way round). – celtschk Jun 14 '14 at 12:42

The trick with the two-dimensional harmonic oscillator is to recognize that there are two directions so that movement in one direction is independent of the movement in the other (if the harmonic oscillator is rotationally symmetric, any two orthogonal directions will do). If you plot the equipotential lines of the oscillator potential (that is, the potential energy if the mass is at that point), it consists of ellipses; the main axes of those ellipses give those two directions.

In each of the directions, the equation of motion is just the equation of motion of a one-dimensional harmonic oscillator. So you solve the two one-dimensional harmonic oscillators separately.

If you don't want to use such a shortcut, you can also calculate it directly using any of the usual methods, like Lagrange formalism or Hamilton formalism.

Here's how you would do it in Lagrange formalism:

*Step 1:* determine the kinetic and potential energy of the 2D harmonic oscillator.

Kinetic energy: $T = \tfrac<1><2>m(\dot x^2+\dot y^2)$

Here $x$ and $y$ are the coordinates, and the dot describes the time derivative, that is, $\dot x$ and $\dot y$ are the components of the velocity.

Potential energy: $V = ax^2 + bxy + cy^2$

Here $a$, $b$ and $c$ are general constants (with the restriction that $a>0$, $c>0$ and $2ac-b^2>0$). This is the most general two-dimensional harmonic oscillator potential with the restriction that the minimum is at $x=y=0$ (and the value there is $0$, but a constant term in the potential doesn't change the equations of motion).

*Step 2:* From the kinetic and potential energy, you calculate the Lagrange function. That step is trivial: The Lagrange function is always $L=T-V$, that is in this case, $$L = \tfrac12m(\dot x^2+\dot y^2) - ax^2-bxy - cy^2$$

*Step 3:* To derive the equation of motion, you just plug this lagrange equation into the Euler-Lagrange equatons (of the second type): For each coordinate $q$ (that is here, $x$ and $y$), the equation of motion reads $$\frac<\mathrm d><\mathrm dt>\frac<\partial L><\partial\dot q> = \frac<\partial L><\partial q>$$ So for $x$, we get $\partial L/\partial\dot x = m\dot x$ and $\partial L/\partial x = 2ax + by$, and thus $$m\ddot x = -(2ax+by)$$ and analogously $$m\ddot y = -(bx + 2cy)$$

Those are the equations of motion.

$x$ and $y$ are just two directions in which the oscillator can be displaced. They don't even need to be orthogonal. The potential energy $ax^2+bxy+cy^2$ is just the most general possible quadratic form without linear or constant term. It has that form because that's what makes the system a harmonic oscillator (you could plug another potential in here, but then you'd no longer have a harmonic oscillator — well, unless your coordinates are something else than displacements, then it may be a harmonic oscillator in different coordinates). – celtschk Jun 14 '14 at 13:22

@math12: "Two-dimensional harmonic oscillator" means, *by definition,* "two-dimensional system with attractive quadratic potential". "Quadratic potential" means "potential that is a polynomial of degree 2 in the position", and "attractive" means "when you go away from the equilibrium position, the potential gets greater". – celtschk Jun 14 '14 at 13:41

The 1D Harmonic Oscillator

The *harmonic oscillator is an extremely important physics problem*. Many potentials look like a harmonic oscillator near their minimum. This is the first non-constant potential for which we will solve the Schrödinger Equation.

The harmonic oscillator Hamiltonian is given by

which makes the *Schrödinger Equation for energy eigenstates*

Note that this potential also has a Parity symmetry. The potential is unphysical because it does not go to zero at infinity, however, it is often a very good approximation, and this potential can be solved exactly.

It is standard to remove the spring constant from the Hamiltonian, replacing it with the *classical oscillator frequency*.

The *Harmonic Oscillator Hamiltonian* becomes.

The *differential equation to be solved is*

To solve the Harmonic Oscillator equation, we will first change to dimensionless variables, then find the form of the solution for . then multiply that solution by a polynomial, derive a recursion relation between the coefficients of the polynomial, show that the polynomial series must terminate if the solutions are to be normalizable, derive the energy eigenvalues, then finally derive the functions that are solutions.

The *energy eigenvalues* are

for . There are a countably infinite number of solutions with *equal energy spacing*. We have been forced to have quantized energies by the requirement that the wave functions be normalizable.

The *ground state wave function* is.

This is a Gaussian (minimum uncertainty) distribution. Since the HO potential has a parity symmetry, the *solutions either have even or odd parity*. The ground state is even parity.

The *first excited state* is an odd parity state, with a first order polynomial multiplying the same Gaussian.

The *second excited state* is even parity, with a second order polynomial multiplying the same Gaussian.

Note that is equal to the number of zeros of the wavefunction. This is a common trend. With more zeros, a wavefunction has more curvature and hence more kinetic energy.

The general solution can be written as

with the coefficients determined by the recursion relation

and the dimensionless variable given by.

Simple Harmonic Oscillator Equation

where and are constants. Moreover, this solution describes a type of oscillation characterized by a constant amplitude, . and a constant angular frequency, . The phase angle, . determines the times at which the oscillation attains its maximum value. The frequency of the oscillation (in hertz) is . and the period is . The frequency and period of the oscillation are both determined by the constant . which appears in the simple harmonic oscillator equation, whereas the amplitude, . and phase angle, . are determined by the initial conditions. [See Equations (10 )-(13 ).] In fact, and are the two arbitrary constants of integration of the second-order ordinary differential equation (17 ). Recall, from standard differential equation theory (Riley 1974), that the most general solution of an th-order ordinary differential equation (i.e. an equation involving a single independent variable, and a single dependent variable, in which the highest derivative of the dependent with respect to the independent variable is th-order, and the lowest zeroth-order) involves arbitrary constants of integration. (Essentially, this is because we have to integrate the equation times with respect to the independent variable to reduce it to zeroth-order, and so obtain the solution. Furthermore, each integration introduces an arbitrary constant. For example, the integral of . where is a known constant, is . where is an arbitrary constant.)

According to Equation (21 ), is a conserved quantity. In other words, it does not vary with time. This quantity is generally proportional to the overall energy of the system. For instance, would be the energy divided by the mass in the mass-spring system discussed in Section 2.1. The quantity is either zero or positive, because neither of the terms on the right-hand side of Equation (22 ) can be negative.

Let us search for an equilibrium state. Such a state is characterized by . so that . It follows from Equation (17 ) that . and from Equation (22 ) that . We conclude that the system can only remain permanently at rest when . Conversely, the system can never permanently come to rest when . and must, therefore, keep moving for ever. Because the equilibrium state is characterized by . we deduce that represents a kind of ``displacement'' of the system from this state. It is also apparent, from Equation (22 ), that attains it maximum value when . In fact,

where is the amplitude of the oscillation. Likewise, attains its maximum value,

The simple harmonic oscillation specified by Equation (18 ) can also be written in the form

where and . Here, we have employed the trigonometric identity . (See Appendix B .) Alternatively, Equation (18 ) can be written

where . and use has been made of the trigonometric identity . (See Appendix B ). It follows that there are many different ways of representing a simple harmonic oscillation, but they all involve linear combinations of sine and cosine functions whose arguments take the form . where is some constant. However, irrespective of its form, a general solution to the simple harmonic oscillator equation must always contain two arbitrary constants. For example, and in Equation (25 ), or and in Equation (26 ).

The simple harmonic oscillator equation, (17 ), is a linear differential equation, which means that if is a solution then so is . where is an arbitrary constant. This can be verified by multiplying the equation by . and then making use of the fact that . Linear differential equations have the very important and useful property that their solutions are *superposable* . This means that if is a solution to Equation (17 ), so that

and is a different solution, so that

then is also a solution. This can be verified by adding the previous two equations, and making use of the fact that . Furthermore, it can be demonstrated that any linear combination of and . such as . where and are constants, is also a solution. It is very helpful to know this fact. For instance, the special solution to the simple harmonic oscillator equation, (17 ), with the simple initial conditions and can be shown to be

Likewise, the special solution with the simple initial conditions and is

Harmonic motion is one of the most important examples of motion in all of physics. Any vibration with a restoring force equal to Hooke’s law is generally caused by a simple harmonic oscillator. The potential for the harmonic ocillator is the natural solution every potential with small oscillations at the minimum. Almost all potentials in nature have small oscillations at the minimum, including many systems studied in quantum mechanics. Here, harmonic motion plays a fundamental role as a stepping stone in more rigorous applications.

The Harmonic Oscillator is characterized by the its Schr ö dinger Equation. This equation is presented in section 1.1 of this manual. The harmonic oscillator has only discrete energy states as is true of the one-dimensional particle in a box problem. The equation for these states is derived in section 1.2. An exact solution to the harmonic oscillator problem is not only possible, but also relatively easy to compute given the proper tools. It is one of the first applications of quantum mechanics taught at an introductory quantum level. Systems with nearly unsolvable equations are often broken down into smaller systems. The solution to this simple system can then be used on them. A firm understanding of the principles governing the harmonic oscillator is prerequisite to any substantial study of quantum mechanics.

The classical potential for a harmonic oscillator is derivable from Hooke’s law. It is conventionally written:

The Summation Substitution. Why is replacing n with (n+2) in the power series derivation mathematically legal?

Most textbooks do not expand on the rational for this substitution. The substitution makes perfect sense, however when each term of the summation is expanded and the derivatives for each term are taken. The steps below might help with the logic behind this part of the derivation of the solution to the harmonic oscillator equation: We are given:

Rewriting this summation in terms of its expansion:

Then taking the first and second derivatives of the expanded terms, we have:

This expansion shows that the first two terms in the second derivative (equation 27 above) are zero because the coefficients are zero. The summation for the second derivative actually begins with n+2. Hence the substitution in the power series derivation above.

Hermite Polynomials. What are Hermite polynomials?

The Hermite polynomial is defined as the solution to Hermite’s Differential equation. This polynomial is a direct result of solving the quantum harmonic oscillator differential equation. The Hermite’s Differential equation takes the familiar form:

Where n is a real, non-negative number (n = 0, 1, 2, 3 )

Hermite polynomials form a complete orthogonal set on the interval - to + with respect to the function e - y 2. The orthogonality relationship can be shown as such 4.

With the orthogonality condition met, piecewise continuous function such as the solution to the quantum harmonic oscillator can be expressed in terms of the equation for Hermite polynomials:

[1] Dicke, Robert H. & Wittke, James P. Introduction to Quantum Mechanics. 1960, Addison Wesley, San Francisco, CA.

[2] Gasiorowicz, Stephen, Quantum Physics. 1974, John Wiley & Sons, New York.

[3] Liboff, Richard L. Introductory Quantum Mechanics. 2003, Addison Wesley, San Francisco, CA.

[4] Saxon David S. Elementary Quantum Mechanics. 1968, Holden-Day, San Francisco, CA.

[5] Schiff, Leonard I. Quantum Mechanics. 1955, McGraw-Hill Book Company, New York.

Books that provide practice problems concerning the harmonic oscillator

[6] Fl ü gge, S. Practical Quantum Mechanics. 1974, Springer-Verlag, New York.

[7] Gol’dman, I. I. & Krivchenkov, V. D. Problems in Quantum Mechanics. 1961, Pergamon Press, Addison-Wesley Publishing Co. Reading, MA.

Websites: For information about Hermite Polynomials:

[10] http://www.sciencedaily.com/encyclopedia/hermite polynomials

For general information about the quantum harmonic oscillator

[11] http://sps.physics.gatech.edu/Quantum Harmonic Oscillator Lecture.pdf

[12] http://www.sciencedaily.com/encyclopedia/quantum harmonic oscillator