Question

(a) Prove that the first term in the Schrödinger equation, -( ℏ² / 2m )( d²ψ / dx² ), reduces to the kinetic energy of the particle multiplied by the wave function for a freely moving particle, with the wave function given by

Accomplish this by calculating the following quantities. (Use the following as necessary: A, i, k, x, ℏ, and m.)

• dψ/dx =
• d²ψ/dx² =
• [(-ℏ²/2m)d²ψ/dx²]/ψ =
• momentum p =
• K = p²/(2m) =

(b) Repeat the previous part, but for a particle in a box, with the wave function given by

.

Accomplish this by calculating the following quantities. (Use the following as necessary: L, n, x, ℏ, and m.)

• dψ/dx =
• d²ψ/dx² =
• [(-ℏ²/2m)d²ψ/dx²]/ψ =
• p =
• K = p²/(2m) =

Answer 1