Ch 40: One-Dimensional Quantum Mechanics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 40: One-Dimensional Quantum Mechanics

Problem 42b

Chapter 40, Problem 42b

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. At what value or values of x is the particle most likely to be found?

Verified step by step guidance

Step 1: Understand the problem. The wave function ψ(x) represents the probability amplitude of finding the particle at position x. The probability density is proportional to |ψ(x)|², which is the square of the wave function's magnitude. To find the most likely position, we need to locate the maximum of |ψ(x)|².

Step 2: Write the expression for the probability density. Since ψ(x) = Ax exp(−x²/a²), the probability density is given by |ψ(x)|² = (Ax exp(−x²/a²))² = A²x² exp(−2x²/a²).

Step 3: To find the maximum probability density, take the derivative of |ψ(x)|² with respect to x and set it equal to zero. This will give the critical points. Use the chain rule and product rule for differentiation: d/dx [A²x² exp(−2x²/a²)] = 0.

Step 4: Solve the derivative equation. After differentiating, you will get an equation involving x. Simplify the equation and solve for x to find the critical points. These are the values of x where the probability density could be maximum or minimum.

Step 5: Analyze the critical points. Use the second derivative test or inspect the behavior of the function to determine which critical points correspond to maxima. These values of x are where the particle is most likely to be found.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 absolute value of the wave function, |ψ(x)|², gives the probability density of the particle's position.

Probability Density

Probability density is a measure that describes the likelihood of finding a particle in a specific region of space. For a given wave function ψ(x), the probability density is calculated as |ψ(x)|². In this context, identifying the maximum value of the probability density helps determine where the particle is most likely to be found.

Maximizing the Probability Density

To find the position where the particle is most likely to be found, one must maximize the probability density |ψ(x)|². This involves taking the derivative of the probability density with respect to x, setting it to zero, and solving for x. The resulting values indicate the positions where the likelihood of finding the particle is highest.

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