The wavefunction of a particle is given by ψ(x)=A cos(k x)+B sin(k x) where A, B, and k are cons

step 1 Okay, so the way function of a particle is given by side of x is equal to a cosine of kx plus b

The wavefunction of a particle is given by ψ(x)=A cos(k x)+B...

The wavefunction of a particle is given by
$$
\psi(x)=A \cos (k x)+B \sin (k x)
$$
where $A, B,$ and $k$ are constants. Show that $\psi$ is a solution of the Schrödinger equation (Eq. 6.13), assuming the particle is free $(U=0),$ and find the corresponding energy $E$ of the particle.

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Key Concepts

Energy Eigenvalue Determination

By substituting a sinusoidal wavefunction into the free particle Schrödinger equation, the relationship between the wave number (k) and the energy eigenvalue is established. This relation demonstrates that the energy is quantized in terms of ?, k, and the mass, typically yielding an energy of the form E = ?²k²/(2m) for a free particle.

Wavefunctions as Linear Combinations

Quantum mechanics allows the general solution of the Schrödinger equation to be a linear combination of basis functions. For a free particle in one dimension, sine and cosine functions are the natural basis functions due to their periodicity and their property of being eigenfunctions of the second derivative operator.

Time-Independent Schrödinger Equation

This differential equation governs the spatial part of the quantum state for systems with time-independent potentials. It expresses the relation between the second derivative of the wavefunction and the energy of the state, forming an eigenvalue problem for the Hamiltonian operator.

Free Particle

A free particle is one that experiences no potential energy (U = 0). In this case, the Schrödinger equation reduces to an association between the kinetic energy operator (involving the second derivative) and the energy eigenvalue, leading to solutions that are sinusoidal in nature.

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