Question

The wave function of a quantum particle of mass m is 𝜓 (𝑥) = 𝐴 cos(𝑘𝑥) + 𝐵 sin(𝑘𝑥) where 𝐴, 𝐵 𝑎𝑛𝑑 𝑘 are constants. (a) Assuming the particle is free (𝑈 = 0), show that 𝜓 (𝑥) is a solution of the 𝑆𝑐ℎ𝑟ö𝑑𝑖𝑛𝑔𝑒𝑟 equation. (b) Find the corresponding energy E of the particle.

Answer 1

The wave function of a quantum particle satisfies the Schrodinger equation

$$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+U(x)\psi(x)=E\psi(x)$$

In the case of free particle

$$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}=E\psi(x)$$ $$\frac{d^2\psi(x)}{dx^2}=\frac{d^2}{dx^2}\left(A\cos(kx)+B\sin(kx)\right)\\ =-k^2\left(A\cos(kx)+B\sin(kx)\right)=-k^2\psi(x)$$

Hence, the $\psi(x)$ is a solution of the Schrodinger equation if

$$E=\frac{\hbar^2 k^2}{2m}$$