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Chapter 7: Problem 2

Can the magnitude of a wave function \((\Psi *(x, t) \Psi(x, t))\) be a negative number? Explain.

Short Answer

Expert verified
No, the magnitude of a wave function (\(\Psi^*(x, t) \Psi(x, t)\)) cannot be a negative number. This is because the probability density is calculated by multiplying the wave function by its complex conjugate, resulting in the product of two real numbers. The product of two real numbers is always positive or zero, so the magnitude of the wave function can never be negative.

Step by step solution

01

Understanding the wave function Ψ(x,t)

The wave function, denoted by Ψ(x,t), is a complex-valued function that describes the probability amplitude of a particle's position and time. It contains all the information about the particle's behavior and encodes the probabilities of the particle's various possible locations.
02

Computing probability density

To find the probability density of a wave function, we need to compute the product of the wave function itself, Ψ(x,t), and its complex conjugate, denoted by Ψ*(x,t). The probability density of the wave function, which represents the probability of finding the particle in a particular position, is given by: Probability Density = \(|\Psi(x,t)|^2 = \Psi^*(x, t) \Psi(x, t)\)
03

Analyzing the probability density

The probability density is calculated by multiplying the wave function by its complex conjugate. This causes all imaginary parts in the complex conjugate to negate the imaginary parts in the wave function, leaving only real numbers. Since we're multiplying two real numbers, the result will either be positive or zero.
04

Concluding if the magnitude of the wave function can be negative

As we saw in Step 3, the magnitude of the wave function, which is the probability density, is obtained by multiplying the wave function and its complex conjugate, both of which are real numbers. Since the product of two real numbers is always positive or zero, the magnitude of a wave function (\(\Psi^*(x, t) \Psi(x, t)\)) cannot be a negative number.

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