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What are the dimensions of a wavefunction that describe a particle free to move in both x and y directions?

What are the dimensions of a wavefunction that describe a particle free to move in both x and y directions?

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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**Question:** What are the dimensions of a wavefunction that describe a particle free to move in both x and y directions? **Answer:** In quantum mechanics, the wavefunction is a mathematical function used to describe the quantum state of a particle and how it behaves in space and time. For a free particle moving in both the x and y directions, the wavefunction would be represented as Ψ(x, y). The dimensions of the wavefunction in this context are not physical dimensions like length or time, but rather, they are associated with the probability amplitude of the particle's position. The wavefunction provides information on the probability density of finding a particle at a particular location in space. The physical dimensions of the wavefunction are typically given in such a way that the probability density, |Ψ(x, y)|², is a dimensionless quantity. Thus, if integrating the probability density over all space equates to a dimensionless probability (i.e., 1), the wavefunction itself must have dimensions of [length]^-1 for two-dimensional space. This ensures that the integral over the area with |Ψ(x, y)|² results in a unitless number, representing a probability. Therefore, the dimensions of Ψ(x, y) for a particle free to move in both x and y directions are [L]^-1, where [L] represents units of length, commonly meters in the International System of Units (SI).
Transcribed Image Text:**Question:** What are the dimensions of a wavefunction that describe a particle free to move in both x and y directions? **Answer:** In quantum mechanics, the wavefunction is a mathematical function used to describe the quantum state of a particle and how it behaves in space and time. For a free particle moving in both the x and y directions, the wavefunction would be represented as Ψ(x, y). The dimensions of the wavefunction in this context are not physical dimensions like length or time, but rather, they are associated with the probability amplitude of the particle's position. The wavefunction provides information on the probability density of finding a particle at a particular location in space. The physical dimensions of the wavefunction are typically given in such a way that the probability density, |Ψ(x, y)|², is a dimensionless quantity. Thus, if integrating the probability density over all space equates to a dimensionless probability (i.e., 1), the wavefunction itself must have dimensions of [length]^-1 for two-dimensional space. This ensures that the integral over the area with |Ψ(x, y)|² results in a unitless number, representing a probability. Therefore, the dimensions of Ψ(x, y) for a particle free to move in both x and y directions are [L]^-1, where [L] represents units of length, commonly meters in the International System of Units (SI).
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