Question

Write the Schrödinger equation (3.22) if \(\psi\) is a function of \(x,\) and \(V=\frac{1}{2} m \omega^{2} x^{2}\) (this is a one-dimensional harmonic oscillator). Find the solutions \(\psi_{n}(x)\) and the energy eigenvalues \(E_{n}\). Hints: In Chapter 12 , equation (22.1) and the first equation in \((22.11),\) replace \(x\) by \(\alpha x\) where \(\alpha=\sqrt{m \omega / \hbar} .\) (Don't forget appropriate factors of \(\alpha\) for the \(x\) 's in the denominators of \(D=d / d x\) and \(\psi^{\prime \prime}=d^{2} \psi / d x^{2} .\) ) Compare your results for equation (22.1) with the Schrödinger equation you wrote above to see that they are identical if \(E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega .\) Write the solutions \(\psi_{n}(x)\) of the Schrödinger equation using Chapter 12, equations (22.11) and (22.12).

Short answer

Energy eigenvalues are given by \( E_n = (n + \frac{1}{2}) \hbar \omega \). The solutions \( \psi_{n}(x) \) for the wave functions involve Hermite polynomials and the Gaussian factor.

Step-by-step solution

Write the Schrödinger equation for the Harmonic Oscillator

In one dimension, the time-independent Schrödinger equation is given by \[ -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi = E \psi \] Substituting the potential function for the harmonic oscillator, \[ V(x) = \frac{1}{2} m \omega^2 x^2 \] results in \[ -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi = E \psi \]

Substitute and simplify using \alpha = \sqrt{m \omega / \hbar}

Introduce dimensionless variables using \( \alpha = \sqrt{m \omega / \hbar} \). Therefore, replace \( x \) with \( \alpha x \) in the equation and due to this, \( d / dx \) becomes \( \alpha d / dx \) and \( d^2 / dx^2 \) becomes \( \alpha^2 d^2 / dx^2 \): \[\begin{aligned} - \frac{\hbar^2}{2m} & \alpha^2 \frac{d^2 \psi}{d(\alpha x)^2} + \frac{1}{2} m \omega^2 (\alpha x)^2 \psi = E \psi \end{aligned}\] Simplify to get \[ -\frac{\hbar^2}{2m} \alpha^2 \frac{d^2 \psi}{d x^2} + \frac{1}{2} m \omega^2 \alpha^2 x^2 \psi = E \psi. \]

Substitute \alpha = \sqrt{m \omega / \hbar}

Substitute \( \alpha = \sqrt{m \omega / \hbar} \) back into the equation to simplify: \[ \begin{aligned} -\frac{\hbar^2}{2m} \left( \frac{m \omega}{\hbar} \right) \frac{d^2 \psi}{dx^2} + \frac{1}{2} m \omega^2 \left( \frac{m \omega x^2}{\hbar} \right) \psi = E \psi \end{aligned} \] Simplifying further, we get \[ -\frac{\hbar \omega}{2} \frac{d^2 \psi}{dx^2} + \frac{\hbar \omega}{2} x^2 \psi = E \psi. \]

Find the general form of solutions \psi_n(x) and energy eigenvalues E_n

From the previously derived differential equation, the eigenfunctions and eigenvalues of the one-dimensional harmonic oscillator are given by \[ E_n = \left( n + \frac{1}{2} \right) \hbar \omega \] The normalized wave functions (solutions) \( \psi_n(x) \) are expressed as \[ \psi_{n}(x) = \left(\frac{\alpha}{\sqrt{\pi} 2^n n!}\right)^{\frac{1}{2}} H_n(\alpha x) e^{- \alpha^2 x^2 / 2} \] where \( H_n \) are the Hermite polynomials and \( \alpha = \sqrt{\frac{m \omega}{\hbar}} \).

Key concepts

time-independent Schrödinger equation

The time-independent Schrödinger equation is a fundamental principle in quantum mechanics. It describes how the quantum state of a physical system changes over time and is given by \( -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi = E \psi \).
In this equation:

• \textbackslash(\textbackslash(\textbackslash psi\textbackslash(\textbackslash) is the wave function. 'E' is the energy of the system. 'V(x)' is the potential energy as a function of position 'x'.

For a one-dimensional harmonic oscillator, we replace \( V(x) \) with \( \frac{1}{2} m \omega^2 x^2 \) to solve for the wave functions and energy levels.

harmonic oscillator potential

The harmonic oscillator potential is crucial for understanding vibrations in molecules or quantum systems. The potential energy function is given by \( V(x) = \frac{1}{2} m \omega^2 x^2 \).
Here:

• 'm' is the mass of the particle.
• 'ω' (omega) is the angular frequency.
• 'x' is the position.

When this potential is applied in the Schrödinger equation, it creates a parabolic potential well where the particle can oscillate. This form of potential leads to quantized energy levels.

energy eigenvalues

The energy eigenvalues represent the possible energy levels that a particle in a quantum system can occupy. For the harmonic oscillator, these are found by solving the Schrödinger equation.
The formula is: \( E_n = \left( n + \frac{1}{2} \right) \hbar \omega \)
In this context:

• 'n' is the quantum number (n = 0, 1, 2,...).
• \textbackslash(\textbackslash(\textbackslash hbar\textbackslash(\textbackslash) is the reduced Planck's constant.
• 'ω' is the angular frequency.

These discrete energy levels show that a particle in a quantum harmonic oscillator can only have specific energies, which is a stark contrast to classical mechanics.

wave functions

The wave functions \(\psi_n(x) \) describe the quantum states of the harmonic oscillator. They are solutions to the Schrödinger equation and contain information about the probability of finding a particle at a particular position. For harmonic oscillators, the wave functions are: \( \psi_{n}(x) = \left(\frac{\alpha}{\sqrt{\pi} 2^n n!}\right)^{\frac{1}{2}} H_n(\alpha x) e^{- \alpha^2 x^2 / 2} \).
Here:

• 'n' is the quantum number.
• 'α' is given by \( \alpha=\sqrt{m \omega / \hbar} \).
• 'H_n' are Hermite polynomials.

These wave functions highlight the non-classical distribution of particles in quantum mechanics.

Hermite polynomials

Hermite polynomials, denoted as \( H_n(x) \), are a series of orthogonal polynomials that arise in probability theory and are utilized to solve the quantum harmonic oscillator problem. They play a key role in expressing the wave functions.

• Their recurrence relation is: \( H_n'(x)=2xH_n(x)-2nH_{n-1}(x) \).
• They help define the shape of the wave functions.

For different quantum numbers 'n', Hermite polynomials generate different polynomial functions, leading to the distinct shapes of \(\psi_n(x)\textbackslash,\textbackslash( \) for each energy level.