Question

A particle in a harmonic oscillator potential has the initial wave function \(\Psi(x, 0)=A\left[\psi_{0}(x)+\psi_{1}(x)\right] .\) Normalize \(\Psi(x, 0)\)

Short answer

Answer: The normalization constant A for the given initial wave function is \(\frac{1}{\sqrt{2}}\).

Step-by-step solution

The normalization condition is given by $$\int_{-\infty}^{\infty} |\Psi(x, 0)|^2 dx = 1$$ Here, \(\Psi(x, 0) = A(\psi_0(x) + \psi_1(x))\), so we need to find the constant A. #tag_step2# Title: Square the wave function and simplify

First, let's square the modulus of the wave function: $$|\Psi(x, 0)|^2 = |A(\psi_0(x) + \psi_1(x))|^2 = A^2(|\psi_0(x)|^2 + |\psi_1(x)|^2 + 2\psi_0(x)\psi_1(x))$$ Since \(\psi_0(x)\) and \(\psi_1(x)\) are orthogonal eigenfunctions of the harmonic oscillator, their product will integrate to zero: $$\int_{-\infty}^{\infty} \psi_0(x)\psi_1(x) dx = 0$$ #tag_step3#Title: Apply normalization condition and solve for A

Now, let's apply the normalization condition: $$1 = \int_{-\infty}^{\infty} A^2(|\psi_0(x)|^2 + |\psi_1(x)|^2) dx = A^2\left( \int_{-\infty}^{\infty} |\psi_0(x)|^2 dx + \int_{-\infty}^{\infty} |\psi_1(x)|^2 dx \right)$$ Since each eigenfunction is individually normalized, their integrals are equal to 1: $$1 = A^2(1 + 1) = 2A^2$$ Solving for A, we get: $$A = \frac{1}{\sqrt{2}}$$ #tag_step4#Title: Write down normalized wave function

Now, we can write down the normalized wave function: $$\Psi(x, 0) = \frac{1}{\sqrt{2}}(\psi_0(x) + \psi_1(x))$$ Thus, we have normalized the given initial wave function of a particle in a harmonic oscillator potential.

Key concepts

Harmonic Oscillator Potential

A harmonic oscillator potential in the context of quantum mechanics mimics the classical system of a mass attached to a spring that oscillates around an equilibrium position. The potential energy of such a system is given by the term \( \frac{1}{2}kx^2 \), where \( k \) is the spring constant and \( x \) is the displacement from equilibrium. This quadratic dependency on \( x \) creates a parabolic potential well in which the particle can 'oscillate' back and forth.

Quantum mechanically, particles confined in a harmonic oscillator potential have quantized energy levels and their behavior is described by wave functions. These wave functions are solutions to the Schrödinger equation for the harmonic oscillator potential. The energy levels are equally spaced, and each level is associated with a distinct wave function, also known as an eigenfunction.

Eigenfunctions of Harmonic Oscillator

The eigenfunctions of a quantum harmonic oscillator are the mathematically derived solutions of the wave functions that correspond to the discrete energy levels of the system. These eigenfunctions are referred to as \( \psi_n(x) \), where \( n \) is the quantum number that denotes different energy levels.

The first few eigenfunctions—including the ground state \( \psi_0(x) \) and the first excited state \( \psi_1(x) \)—are particularly important since they form the basis for understanding more complex quantum states. It's essential to note that the eigenfunctions of the harmonic oscillator are not just any functions; they satisfy specific criteria. They are orthogonal, which means their overlap integral is zero unless they are the same function, and they are normalized, implying that the integral of their square is equal to one. These properties are crucial when dealing with combinations or superpositions of eigenfunctions.

Quantum Mechanics

Quantum Mechanics is the fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It introduces concepts that can seem non-intuitive, such as wave-particle duality, quantization, and superposition. Unlike classical mechanics, quantum mechanics does not provide a deterministic outcome but rather probabilities of finding a particle in a particular state.

In quantum mechanics, the wave function plays a central role as it contains all the information about a quantum system. By solving the Schrödinger equation for a given potential, one obtains the wave functions (and associated energies) that describe the possible states of a system. Understanding wave functions and their behaviors under different potentials, like the harmonic oscillator potential, is a key part of any study within this scientific domain.

Normalization Condition

The normalization condition in quantum mechanics is a requirement that the probability of finding a particle anywhere in space must be one. Mathematically, this condition is expressed by the integral of the absolute square of the wave function over all space being equal to one, symbolically \( \int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1 \). This integral is essentially a sum (actually, a continuous sum which is an integral) of probabilities across all space.

When a wave function is not normalized, it can be made so by multiplying by a constant factor. This factor is what we solve for when normalizing a wave function like the combination of two eigenfunctions of the harmonic oscillator. By ensuring that the wave functions are normalized, we can use them to make meaningful physical predictions about the behavior of quantum systems.