# QM 1-D Harmonic Oscillator Eigenfunction Problem

• tis
In summary, the constant \lambda in the second excited state of a 1-D Harmonic oscillator potential with frequency \omega is not arbitrary and can be determined by substituting into the time-independent Schrodinger Equation and using the orthogonality condition for wavefunctions. This results in \lambda = -1.
tis

## Homework Statement

A particle of mass $m$ moves in a 1-D Harmonic oscillator potential with frequency $\omega.$
The second excited state is $\psi_{2}(x) = C(2 \alpha^{2} x^{2} + \lambda) e^{-\frac{1}{2} a^{2} x^{2}}$ with energy eigenvalue $E_{2} = \frac{5}{2} \hbar \omega$.

$C$ and $\lambda$ are constants. Show that the constant $\lambda$ is not arbitrary.

NOTE: $\int_{-\infty}^{\infty} e^{-\frac{1}{2} a x^{2}} dx = \sqrt{2\pi / a}, a > 0$.

## Homework Equations

That I can think of: TISE, normalization condition for wavefunctions.

## The Attempt at a Solution

I assume there's a solution in substituting into the time-independent Schrodinger Equation and solving for $\lambda$, but the equation seems very difficult. And I have no idea how to utilize the hint provided; everything I've tried (normalization condition, etc.) just gives a dependence on C. I've scoured my textbooks with no luck, their only relevant info is using ladder operators and other methods to produce eigenfunctions.

Just after a point in the right direction. Thanks in advance for your help!

Your idea to substitute this into the TISE seems like a good start. All you will have to do is take derivatives and rearrange some terms, and you will end up with an equation which determines lambda. Give it a try.

Figured it out. Take the ground state of the harmonic oscillator $\psi_{0} = A e^{-\frac{1}{2}\alpha^{2} x^{2}}$ and use the orthogonality condition $\int_{-\infty}^{\infty} \psi_{0}^{*} \psi_{2} \ dx = 0$.

From there just expand, cancel out A* and C, and solve the integrals with integration by parts and the hint provided. Eventually you come to $0=2\alpha^{2}\frac{\sqrt{\pi}}{2\alpha^{3}}+\lambda\frac{\sqrt{\pi}}{ \alpha }$, hence $\lambda=-1$.

## 1. What is the QM 1-D Harmonic Oscillator Eigenfunction Problem?

The QM 1-D Harmonic Oscillator Eigenfunction Problem is a fundamental problem in quantum mechanics that involves finding the wave function (eigenfunction) that describes the behavior of a particle in a one-dimensional harmonic oscillator potential. This problem is important because it allows us to understand the behavior of quantum systems, such as atoms and molecules, which are described by the principles of quantum mechanics.

## 2. What is a harmonic oscillator potential?

A harmonic oscillator potential is a mathematical model that describes the potential energy of a system as a function of the position of a particle. It is characterized by a parabolic shape, with the minimum of the potential being at the equilibrium position. This type of potential is commonly used to model the behavior of systems in which a particle is subject to a restoring force, such as a spring or a pendulum.

## 3. What is an eigenfunction?

An eigenfunction is a special type of function that describes the state of a quantum system. In the context of the QM 1-D Harmonic Oscillator Eigenfunction Problem, the eigenfunctions are solutions to the Schrödinger equation that describe the probability of finding a particle at a specific position in the harmonic oscillator potential. These eigenfunctions are also known as wave functions.

## 4. How is the QM 1-D Harmonic Oscillator Eigenfunction Problem solved?

The QM 1-D Harmonic Oscillator Eigenfunction Problem is solved by using mathematical techniques, such as the method of separation of variables or the ladder operator method. These methods allow us to find the eigenfunctions and corresponding eigenvalues, which determine the energy levels of the harmonic oscillator. The solutions can also be verified by using numerical methods, such as the finite difference method.

## 5. Why is the QM 1-D Harmonic Oscillator Eigenfunction Problem important?

The QM 1-D Harmonic Oscillator Eigenfunction Problem is important because it is a fundamental problem in quantum mechanics that helps us understand the behavior of quantum systems. The solutions to this problem also have practical applications in fields such as physics, chemistry, and engineering. Additionally, the QM 1-D Harmonic Oscillator Eigenfunction Problem serves as a starting point for more complex problems in quantum mechanics and helps lay the foundation for further research in the field.

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