Quantum Mechanics

To determine the eigenvalue of the operator

%5Cfrac%7Bd%5E2%7D%7Bdx%5E2%7D

, we first apply it to the wave function

%5Cpsi%20%3D%20A%20e%5E%7Bikx%7D%20%2B%20B%20e%5E%7B-ikx%7D

Step 1: Apply the operator $%5Cfrac%7Bd%5E2%7D%7Bdx%5E2%7D$ to the wave function. First, compute the first derivative of $%5Cpsi$ : $%5Cfrac%7Bd%7D%7Bdx%7D%20(A%20e%5E%7Bikx%7D%20%2B%20B%20e%5E%7B-ikx%7D)%20%3D%20A%20ik%20e%5E%7Bikx%7D%20-%20B%20ik%20e%5E%7B-ikx%7D$ Next, compute the second derivative: $%5Cfrac%7Bd%5E2%7D%7Bdx%5E2%7D%20(A%20e%5E%7Bikx%7D%20%2B%20B%20e%5E%7B-ikx%7D)%20%3D%20-%20A%20k%5E2%20e%5E%7Bikx%7D%20-%20B%20k%5E2%20e%5E%7B-ikx%7D$ $%3D%20-%20k%5E2%20(A%20e%5E%7Bikx%7D%20%2B%20B%20e%5E%7B-ikx%7D)$

Step 2: Interpret the result. We have $%5Cfrac%7Bd%5E2%7D%7Bdx%5E2%7D%20%5Cpsi%20%3D%20-%20k%5E2%20%5Cpsi$ . This shows that $%5Cpsi$ is an eigenfunction with eigenvalue $-k%5E2$ . Thus, the correct answer is $-k%5E2$ .

Final Answer: $%5Cboxed%7B%5Ctext%7B(2)%20%7D%20-k%5E2%7D$

For a particle in a three-dimensional cubic box, the energy levels are given by the formula:

E_%7Bn_x%2C%20n_y%2C%20n_z%7D%20%3D%20%5Cfrac%7Bh%5E2%7D%7B8ma%5E2%7D%20%5Cleft(%20n_x%5E2%20%2B%20n_y%5E2%20%2B%20n_z%5E2%20%5Cright)

Where

n_x%2C%20n_y%2C%20n_z

are the quantum numbers in the x, y, and z directions, and

a

is the length of the edge of the cubic box. The degeneracy of an energy level is determined by the number of different sets of quantum numbers

(n_x%2C%20n_y%2C%20n_z)

that lead to the same energy. For the given energy

%5Cfrac%7B14h%5E2%7D%7B8ma%5E2%7D

, the degeneracy is 6, as there are 6 distinct combinations of quantum numbers that lead to this energy level. Final Answer:

%5Cboxed%7B6%7D

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Quantum Mechanics

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs