Question

Is the following statement true or false? The larger the amplitude of a Schrödinger wave function, the larger its kinetic energy. Explain your answer.

Short answer

Answer: Yes, it is true that the larger the amplitude of a Schrödinger wave function, the larger its kinetic energy, as demonstrated through the comparison of two wave functions with different amplitudes but equal forms and energy. The kinetic energy of the respective wave functions shows a direct proportionality to their amplitudes.

Step-by-step solution

Understand the Schrödinger Wave Function

The Schrödinger wave function, denoted as ψ(x), is a mathematical function that describes the probability distribution of a particle in a quantum system. The square of the wave function's magnitude, |ψ(x)|^2, represents the probability density of finding the particle at a given position x.

Calculate the Kinetic Energy Operator

Kinetic energy in a quantum system is calculated using the kinetic energy operator, which is given by: - \(\frac{ħ^2}{2m} \frac{d^2}{dx^2}\) Here, ħ is the reduced Planck's constant, m is the mass of the particle, and the second derivative term represents the spatial derivative of the wave function with respect to position x.

Consider a Wave Function with Different Amplitudes

Suppose we have two wave functions ψ_1(x) and ψ_2(x) that describe the same spatial distribution of a particle with equal forms and equal energy but with different amplitudes A_1 and A_2, where A_2 > A_1. This means that ψ_2(x) = A_2 * ψ_1(x).

Calculate the Kinetic Energy of the Respective Wave Functions

In order to find the kinetic energy of each wave function, we need to act the kinetic energy operator on each wave function respectively. We have ψ_1(x) and ψ_2(x) = A_2 * ψ_1(x), so the kinetic energy operator acting on each wave function gives: K.E._1 = -\(\frac{ħ^2}{2m}\) * \(\frac{d^2ψ_1(x)}{dx^2}\) K.E._2 = -\(\frac{ħ^2}{2m}\) * \(\frac{d^2(A_2 * ψ_1(x))}{dx^2}\) By using the linearity of the operator, this simplifies to: K.E._2 = -\(\frac{ħ^2}{2m}\) * (A_2 * \(\frac{d^2ψ_1(x)}{dx^2}\))

Compare the Kinetic Energy of the Respective Wave Functions

Comparing the kinetic energies, we can see that: K.E._2 = A_2 * K.E._1 This means that the kinetic energy of the second wave function is directly proportional to the amplitude of the first wave function. Therefore, as the amplitude of the Schrödinger wave function increases, so does the kinetic energy of the particle. In conclusion, the statement is true: "The larger the amplitude of a Schrödinger wave function, the larger its kinetic energy."