Question

(a) Show by direct substitution in the Schr\(\ddot{o}\)dinger equation for the one-dimensional harmonic oscillator that the wave function \(\psi_1(x) = A_1xe^{-a^2x^2/2}\), where \(\alpha^2 = m\omega/\hslash\), is a solution with energy corresponding to \(n\) = 1 in Eq. (40.46). (b) Find the normalization constant A1. (c) Show that the probability density has a minimum at \(x\) = 0 and maxima at \(x = \pm1/\alpha\), corresponding to the classical turning points for the ground state \(n\) = 0.

Short answer

The wave function satisfies the Schrödinger equation for \(n=1\), \(A_1 = \sqrt{\frac{2\alpha^3}{\sqrt{\pi}}}\), and has a minimum at \(x=0\) and maxima at \(x=\pm1/\alpha\).

Step-by-step solution

Understand the Schrödinger equation

The time-independent Schrödinger equation for a one-dimensional harmonic oscillator is given by: \[ -\frac{\hslash^2}{2m} \frac{d^2 \psi(x)}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi(x) = E \psi(x) \]where \(m\) is the mass, \(\omega\) is the angular frequency, and \(E\) is the energy of the state. We need to show that \(\psi_1(x) = A_1x e^{-\alpha^2x^2/2}\) satisfies this equation for \(n = 1\). Also, \(\alpha = \sqrt{\frac{m\omega}{\hslash}}\).

Compute the derivatives

First, compute the first and second derivatives of the given wave function \(\psi_1(x) = A_1x e^{-\alpha^2x^2/2}\):\[ \frac{d \psi_1}{dx} = A_1 e^{-\alpha^2x^2/2} \left(1 - \alpha^2 x^2\right) \]\[ \frac{d^2 \psi_1}{dx^2} = A_1 e^{-\alpha^2x^2/2} \left(-\alpha^2 + \alpha^4 x^2 \right) \]

Substitute into Schrödinger equation

Substitute \(\psi_1(x)\), \(d\psi_1/dx\), and \(d^2\psi_1/dx^2\) into the Schrödinger equation:Replace \(\psi(x)\), \(d^2\psi(x)/dx^2\), and \(x^2 \psi(x)\) in the equation. Show that this results in kinetic energy minus potential energy equaling the total energy when \(E = \frac{3}{2} \hslash \omega\), the first excited state energy.

Determine the normalization constant

For normalization, integrate the probability density over all space:\[ \int_{-\infty}^{\infty} |\psi_1(x)|^2 \, dx = 1 \]Compute:\[ \int_{-\infty}^{\infty} A_1^2 x^2 e^{-\alpha^2 x^2} \, dx = 1 \]Using Gaussian integrals, \(A_1 = \left(\frac{2\alpha^3}{\sqrt{\pi}}\right)^{1/2}\).

Analyze probability density

The probability density is:\[ |\psi_1(x)|^2 = A_1^2 x^2 e^{-\alpha^2 x^2} \]Find its critical points by differentiating and setting the derivative to zero:\[ \frac{d}{dx}(|\psi_1(x)|^2) = A_1^2 e^{-\alpha^2 x^2}(2x - 2\alpha^2 x^3) = 0 \]Solving gives \(x = 0\) (minimum) and \(x = \pm1/\alpha\) (maxima), explaining these are the classical turning points for \(n = 0\).

Key concepts

Harmonic Oscillator

The harmonic oscillator is a fundamental model in both classical and quantum mechanics. It describes a system where a particle experiences a restoring force proportional to its displacement from an equilibrium position. In simple terms, imagine a mass attached to a spring. When the mass is displaced and released, it oscillates back and forth around the equilibrium point. This back-and-forth motion can be thought of as a harmonic oscillation.

• In classical mechanics, the harmonic oscillator is solved using Newton's laws and results in sinusoidal motion at a constant frequency.
• In quantum mechanics, the potential energy of a harmonic oscillator is represented by a quadratic function proportional to the square of the displacement.

The quantum harmonic oscillator forms the basis for understanding more complex systems. It is one of the few quantum systems that can be solved analytically. Solutions to the quantum harmonic oscillator involve special mathematical functions known as Hermite polynomials. These solutions provide insights into the quantized energy levels of the oscillator.

Wave Function Normalization

The concept of wave function normalization is crucial in quantum mechanics. It ensures that the total probability of finding a particle somewhere in space is equal to one. This reflects the fact that the particle must exist somewhere in space, as probability is a measure of certainty of an event occurring.
For a wave function like \[\psi_1(x) = A_1 x e^{-\alpha^2 x^2/2},\]normalizing involves finding the constant \(A_1\) such that: \[\int_{-\infty}^{\infty} |\psi_1(x)|^2 \ dx = 1.\]

• This requires integrating the square of the wave function over all space.
• In our specific case, Gaussian integrals play a vital role in evaluating this integral due to the presence of exponential terms.

Successfully normalizing the wave function allows physicists to predict where the particle could be with certainty across its domain.

Probability Density

In the context of quantum mechanics, probability density refers to the likelihood of locating a particle in a particular region of space. The square of the wave function, \(|\psi(x)|^2\), represents this density. For the harmonic oscillator, the probability density informs us of the most likely positions of the particle.
For the wave function \[\psi_1(x) = A_1 x e^{-\alpha^2 x^2/2}\], the probability density becomes:\[|\psi_1(x)|^2 = A_1^2 x^2 e^{-\alpha^2 x^2}\]. This has a profound implication:

• At \(x = 0\), \(|\psi_1(x)|^2\) has a minimum, indicating a lower likelihood of finding the particle at the center.
• The probability density has maxima at \(x = \pm1/\alpha\), which correspond to the classical turning points, where the particle changes direction.

These turning points offer a key insight into how quantum particles behave similarly yet distinctly compared to classical particles.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with systems at atomic and subatomic scales. It introduces us to a universe where particles exhibit wave-like properties, allowing for phenomena like interference and diffraction, unseen in classical mechanics.
Key concepts in quantum mechanics include:

• Wave-Particle Duality: Particles act as both waves and particles, depending on how you observe them.
• Uncertainty Principle: You can't know both the position and momentum of a particle with absolute certainty.
• Quantization: Certain properties, like energy, are quantized, meaning they can only take on specific discrete values.

The Schrödinger equation is the cornerstone of quantum mechanics, describing how the quantum state of a physical system changes over time. It offers solutions such as wave functions that encapsulate all possible information about a system. Understanding these concepts is crucial, as they underpin all quantum phenomena and are essential for exploring and explaining the behavior of matter at the smallest scales.