Question

Is the superposition of two wave functions, which are solutions to the Schrödinger equation for the same potential energy, also a solution to the Schrödinger equation? a) no b) yes c) depends on potential energy d) only if \(\frac{d^{2} \psi(x)}{d x^{2}}=0\)

Short answer

Answer: (b) Yes

Step-by-step solution

Write down the time-independent Schrödinger equation

The time-independent Schrödinger equation is given by: \(-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x)\), where \(\psi(x)\) is the wave function, \(V(x)\) is the potential energy, \(E\) is the energy eigenvalue, \(m\) is the mass of the particle, and \(\hbar\) is the reduced Planck's constant.

Define the superposition of the two wave functions

Let \(\psi_1(x)\) and \(\psi_2(x)\) be the two wave functions that are solutions to the Schrödinger equation for the same potential energy \(V(x)\). Their superposition can be represented as a linear combination: \(\psi(x) = c_1 \psi_1(x) + c_2 \psi_2(x)\), where \(c_1\) and \(c_2\) are some constants.

Substitute the superposition into the Schrödinger equation

Now we will substitute the superposition of the two wave functions (the linear combination) into the Schrödinger equation to check whether it satisfies the equation: \(-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} (c_1\psi_1(x) + c_2\psi_2(x)) + V(x) (c_1\psi_1(x) + c_2\psi_2(x)) = E (c_1\psi_1(x) + c_2\psi_2(x))\)

Check if the superposition satisfies the Schrödinger equation

By taking the second derivative of the linear combination and applying it to the Schrödinger equation, we get: \(-\frac{\hbar^2}{2m} (c_1 \frac{d^2\psi_1(x)}{dx^2} + c_2 \frac{d^2\psi_2(x)}{dx^2}) + V(x) (c_1\psi_1(x) + c_2\psi_2(x)) = E (c_1\psi_1(x) + c_2\psi_2(x))\). Since both \(\psi_1(x)\) and \(\psi_2(x)\) are solutions to the Schrödinger equation, they both satisfy the equation. Therefore, the left-hand side of the above equation equals to the right-hand side, and the superposition of the two wave functions also satisfies the Schrödinger equation. Hence, the superposition of two wave functions is also a solution to the Schrödinger equation. The correct answer is option (b) yes.

Key concepts

Wave Function

In quantum mechanics, the wave function is a fundamental concept that describes the quantum state of a particle or system of particles. It is represented by the symbol \( \psi(x) \) and is a mathematical function that provides information about the probability amplitude of a particle's position and momentum.

The square of the absolute value of the wave function, \( |\psi(x)|^2 \) gives the probability density, that is, the likelihood of finding the particle at a certain position in space. This interpretation is known as the Born rule. An essential feature of the wave function is that it can exhibit superposition, meaning that if \( \psi_1(x) \) and \( \psi_2(x) \) are wave functions, their linear combination \( \psi(x) = c_1 \psi_1(x) + c_2 \psi_2(x) \) for constants \( c_1 \) and \( c_2 \) is also a valid wave function. This superposition principle allows for the combination of multiple states into a new, equally valid quantum state.

Potential Energy

The term potential energy in quantum mechanics is the energy associated with the position of a particle within a potential field, denoted as \( V(x) \). It describes the energy due to an object's position relative to others, stresses within itself, its electric charge, and other factors.

The potential energy is a critical component in the Schrödinger equation, defining how a particle behaves within a force field. In quantum systems, the potential can represent various forces, such as electromagnetic or gravitational. It’s worth noting that the potential energy can also shape the wave function since particles tend to be found in regions of low potential energy, where the probability is higher. When solving the Schrödinger equation, any potential energy function \( V(x) \) specific to the system must be considered to obtain accurate wave functions.

Energy Eigenvalue

The term energy eigenvalue, denoted as \( E \), refers to a characteristic value that can be derived from the Schrödinger equation. In the context of quantum mechanics, eigenvalues correspond to measurable values, such as the energy levels of an atom or molecule.

When a wave function is an eigenfunction of the Hamiltonian operator (which includes kinetic and potential energy terms), the corresponding eigenvalue is a possible outcome for the energy measurement of the system. These discrete energy eigenvalues are also known as quantized energy levels, which is a distinguishing feature of quantum systems. Understanding energy eigenvalues is fundamental for interpreting spectra, chemical bonding, and the stability of matter.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It introduces concepts and principles that differ significantly from classical physics, such as quantization, uncertainty, and superposition.

The Schrödinger equation, which effectively describes how quantum systems evolve, is one of the cornerstones of quantum mechanics. It occupies a role similar to Newton's laws in classical mechanics. Quantum mechanics also predicts that particles like electrons exhibit both wave-like and particle-like properties, a phenomenon known as wave-particle duality. Moreover, it allows particles to exist in multiple states at once, through superposition, until a measurement causes the system to collapse into one of the probable states. Interactions and the role of the observer are also uniquely interpreted in quantum mechanics, leading to continued research and discussions in understanding the nature of reality.