CALC Determine the normalization constant A1 for the n = 1 ground-state wave function of the quantum harmonic oscillator. Your answer will be in terms of b.

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1. The wave function for the ground state (n=1) of a quantum harmonic oscillator is given by \( \psi_1(x) = A_1 e^{-\frac{b^2 x^2}{2}} \), where \( A_1 \) is the normalization constant and \( b \) is a parameter related to the oscillator's properties.
2. To normalize the wave function, we use the condition \( \int_{-\infty}^{\infty} |\psi_1(x)|^2 dx = 1 \). Substituting \( \psi_1(x) \), this becomes \( \int_{-\infty}^{\infty} A_1^2 e^{-b^2 x^2} dx = 1 \).
3. Factor out \( A_1^2 \) since it is constant, leaving \( A_1^2 \int_{-\infty}^{\infty} e^{-b^2 x^2} dx = 1 \). The integral \( \int_{-\infty}^{\infty} e^{-b^2 x^2} dx \) is a standard Gaussian integral, which evaluates to \( \sqrt{\frac{\pi}{b^2}} \).
4. Substitute the result of the Gaussian integral into the normalization condition: \( A_1^2 \sqrt{\frac{\pi}{b^2}} = 1 \).
5. Solve for \( A_1 \) by isolating it: \( A_1 = \sqrt{\frac{b}{\sqrt{\pi}}} \). This is the normalization constant for the ground-state wave function.

Key Concepts

Quantum Harmonic Oscillator

The quantum harmonic oscillator is a fundamental model in quantum mechanics that describes a particle subject to a restoring force proportional to its displacement from an equilibrium position. It is characterized by quantized energy levels and wave functions that are solutions to the Schrödinger equation. The ground state (n=0) and excited states (n=1, 2, ...) represent different energy levels, with the ground state being the lowest energy configuration.

Normalization of Wave Functions

Normalization is a crucial concept in quantum mechanics that ensures the total probability of finding a particle within a given space is equal to one. For a wave function ψ(x), normalization involves calculating a constant A such that the integral of |ψ(x)|² over all space equals one. This process is essential for ensuring that the wave function accurately represents a physical state.

Ground-State Wave Function

The ground-state wave function of the quantum harmonic oscillator describes the lowest energy state of the system. It is typically represented as a Gaussian function, which reflects the probability distribution of the particle's position. The normalization constant A1 is determined through integration, ensuring that the area under the probability density function equals one, thus confirming the wave function's validity.