Consider a wave function given by $\psi \left(x\right)=Asinkx$, where $k=2\pi /\lambda$ and $A$ is a real constant.
(a) For what values of $x$ is there the highest probability of finding the particle described by this wave function? Explain.
(b) For which values of $x$ is the probability zero? Explain.

Verified step by step guidance

1. Step 1: Understand the wave function ψ(x) = A sin(kx). The probability density of finding the particle is proportional to the square of the wave function, |ψ(x)|². Since ψ(x) involves a sine function, its square will depend on the amplitude of the sine wave.
2. Step 2: Recall the properties of the sine function. The sine function, sin(kx), oscillates between -1 and 1. The square of the sine function, sin²(kx), will oscillate between 0 and 1. The highest probability corresponds to the maximum value of sin²(kx), which is 1.
3. Step 3: To find the values of x where the probability is highest, identify where sin²(kx) = 1. This occurs when sin(kx) = ±1. For sin(kx) = ±1, kx must equal (2n+1)π/2, where n is an integer. Substitute k = 2π/λ to express x in terms of λ: x = (2n+1)λ/4.
4. Step 4: To find the values of x where the probability is zero, identify where sin²(kx) = 0. This occurs when sin(kx) = 0. For sin(kx) = 0, kx must equal nπ, where n is an integer. Substitute k = 2π/λ to express x in terms of λ: x = nλ/2.
5. Step 5: Summarize the results. The highest probability of finding the particle occurs at x = (2n+1)λ/4, and the probability is zero at x = nλ/2, where n is an integer. These positions correspond to the maxima and nodes of the sine wave, respectively.

Key Concepts

Wave Function

A 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 state. The square of the absolute value of the wave function, |ψ(x)|², gives the probability density of finding the particle at position x.

Probability Density

Probability density is a measure that describes the likelihood of finding a particle in a specific location in space. For a wave function ψ(x) = A sin(kx), the probability density is given by |ψ(x)|² = A² sin²(kx). The peaks of this function indicate where the particle is most likely to be found, while the zeros indicate locations where the probability of finding the particle is zero.

Nodes and Antinodes

In the context of wave functions, nodes are points where the wave function equals zero, resulting in zero probability of finding the particle. Antinodes, on the other hand, are points where the wave function reaches its maximum value, indicating the highest probability of finding the particle. For the function ψ(x) = A sin(kx), nodes occur at integer multiples of λ/2, while antinodes occur at odd multiples of λ/4.