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Ch 40: One-Dimensional Quantum Mechanics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 40: One-Dimensional Quantum MechanicsProblema 42a
Capitolo 40, Problema 42a

CALC A particle of mass m has the wave function ψ(x) = Ax exp (−x²/a²) when it is in an allowed energy level with E = 0. Draw a graph of ψ(x) versus x.

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Understand the given wave function: ψ(x) = Ax exp(−x²/a²). This is a Gaussian function, which is symmetric about x = 0 and decreases exponentially as x moves away from the origin.
Identify the parameters in the wave function: 'A' is the normalization constant, 'a' determines the width of the Gaussian curve, and 'x' is the position variable.
Recognize that the graph of ψ(x) versus x will have a peak at x = 0 (since exp(−x²/a²) is maximum at x = 0) and will symmetrically decrease on both sides of the origin.
Sketch the graph qualitatively: Start by plotting a bell-shaped curve centered at x = 0. Ensure the curve is symmetric about the y-axis and approaches zero as x → ±∞.
Label the axes: The x-axis represents the position variable 'x', and the y-axis represents the wave function ψ(x). Indicate the peak at x = 0 and the gradual decay of the function as x moves away from the origin.

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Wave Function

The wave function, denoted as ψ(x), describes the quantum state of a particle in quantum mechanics. It contains all the information about the system and is used to calculate probabilities of finding a particle in a particular position. The square of the wave function's absolute value, |ψ(x)|², gives the probability density of the particle's position.
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Intro to Wave Functions

Gaussian Function

The given wave function includes an exponential term exp(−x²/a²), which is a Gaussian function. This function is characterized by its bell-shaped curve, centered at x=0, and it rapidly decreases as x moves away from the center. Gaussian functions are significant in quantum mechanics as they often represent the probability distributions of particles.
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Normalization

Normalization is a crucial concept in quantum mechanics that ensures the total probability of finding a particle in all space equals one. For the wave function ψ(x) to be physically meaningful, it must be normalized, which involves determining the constant A such that the integral of |ψ(x)|² over all space equals one. This process is essential for accurately interpreting the wave function's probabilities.
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The Normal Force
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