Question

True or False: The larger the amplitude of a Schrödinger wave function, the larger its kinetic energy. Explain your answer.

Short answer

Explain your reasoning. Answer: No, a larger amplitude of a Schrödinger wave function does not necessarily mean a larger kinetic energy. The kinetic energy is influenced by the second derivative of the wave function rather than the amplitude.

Step-by-step solution

Review the Schrödinger equation

The time-independent Schrödinger equation is given by: \[ \hat{H} \psi(x) = E \psi(x) \] Where \(\hat{H}\) is the Hamiltonian operator, which describes the total energy of the system, \(E\) is the energy eigenvalue, and \(\psi(x)\) is the wave function. The Hamiltonian operator can be broken down into two components - the kinetic energy operator, \(\hat{T}\), and the potential energy operator, \(\hat{V}\): \[ \hat{H} = \hat{T} + \hat{V} \]

Review the kinetic energy operator

The kinetic energy operator for a particle in one dimension is given by: \[ \hat{T} = -\dfrac{\hbar^2}{2m} \dfrac{d^2}{dx^2} \] Where \(m\) is the particle's mass, and \(\hbar\) is the reduced Planck constant.

Relate the amplitude of the wave function to kinetic energy

The amplitude of the wave function, which is the square root of the probability density, is related to the modulus of the wave function: \[ \lvert \psi(x) \rvert ^2 \] However, the kinetic energy of a particle is not directly related to the wave function's amplitude (or the square root of the probability density). Instead, it is connected to the spatial derivatives of the wave function. We can see from the kinetic energy operator that the second derivative of the wave function influences the kinetic energy.

Determine the validity of the given statement

So, when we consider the Schrödinger equation and its components, we find that the amplitude of the wave function doesn't have a direct impact on the kinetic energy of the particle. The second derivative of the wave function is the primary factor influencing the kinetic energy. Therefore, the given statement is: False. The larger the amplitude of a Schrödinger wave function does not necessarily mean a larger kinetic energy. The kinetic energy is influenced by the second derivative of the wave function rather than the amplitude.

Key concepts

Wave Function

The wave function, represented mathematically as \(\psi(x)\), is a crucial concept in quantum mechanics. It provides a description of the quantum state of a particle. Physically, the wave function determines the probability distribution of a particle's position. This means the square of the absolute value of the wave function, \( |\psi(x)|^2 \), represents the probability density.
It's often said that where \( |\psi(x)|^2 \) is larger, the particle is more likely to be found. However, it does not directly correlate to the particle’s kinetic or total energy. Understanding the role of the wave function aids in visualizing the quantum behavior of particles. Still, one should not equate high amplitude directly with high energy, especially kinetic energy.

Kinetic Energy

Kinetic energy in a quantum system is expressed through the kinetic energy operator, \( \hat{T} \). This operator determines the kinetic energy of a particle based on the wave function's curvature.
Let's look at the operator form: \[ \hat{T} = -\dfrac{\hbar^2}{2m} \dfrac{d^2}{dx^2}\]This formula tells us that the kinetic energy is linked to the second derivative of the wave function.

• \(\hbar\) is the reduced Planck constant.
• \(m\) is the particle's mass.
• The second derivative indicates how the wave function changes curvature-wise.

Therefore, it's the "shape" or "curvature" of the wave function that impacts kinetic energy, not the "height" or amplitude of the wave function itself.

Hamiltonian Operator

The Hamiltonian operator, denoted as \(\hat{H}\), plays a central role in quantum mechanics, particularly in the Schrödinger equation:
\[ \hat{H} \psi(x) = E \psi(x)\]This operator acts on the wave function \(\psi(x)\) and is composed of two main parts:

• Kinetic Energy Operator \(\hat{T}\)
• Potential Energy Operator \(\hat{V}\)

The Hamiltonian provides the system's total energy by summing kinetic and potential energies. In this way, it's a complete descriptor of the particle's dynamics in that quantum state.
It's crucial to remember that while the Hamiltonian interacts with the wave function, it doesn't directly exchange with the function's amplitude but rather derives the energy through operators that reflect the wave function's spatial and potential characteristics.