Quantum Mechanics

Step 1: Understanding the energy levels.
The energy levels for a particle in a three-dimensional box are given by: where are positive integers, and the energy is quantized. The energy given is , so we need to find the combination of that satisfies this energy.
Step 2: Determine the quantum numbers.
We need to find the values of such that: The possible combinations of that satisfy this are: Thus, there are 6 combinations.
Step 3: Conclusion.
Thus, the degeneracy of this energy level is 6.

Step 1: Understanding the potential function.
The potential is given by . To find the minimum energy at which the particle can be bound, we need to analyze the potential's local minima and maxima. Step 2: Determine the critical points.
To find the critical points, take the derivative of the potential . Setting , we get the equation , which gives the critical points at and . Step 3: Minimum energy.
The energy required to escape from the potential well is the energy at the local minima, which is between 0.24 and 0.26 units.

Step 1: Understanding the ground state energy.
For a quantum harmonic oscillator, the energy levels are quantized, and the ground state energy depends on the particle type and statistics. The ground state energy for spin-less fermions is determined by the Pauli exclusion principle, leading to . For electrons, the energy is different due to the spin and statistical properties of fermions, yielding . For bosons, the energy levels are not affected by the Pauli exclusion principle, and their ground state energy is lower, resulting in .
Step 2: Conclusion.
Thus, the correct answers are options (A), (C), and (D).

Step 1: Apply Heisenberg's uncertainty principle.
Heisenberg’s uncertainty principle gives a relation between the uncertainty in position and the uncertainty in momentum : where . Since the proton is confined within a nucleus of size , we can calculate the uncertainty in momentum .
Step 2: Calculate the uncertainty in momentum.
Rearranging the uncertainty principle to solve for : Substitute the known values:
Step 3: Calculate the uncertainty in velocity.
The uncertainty in velocity is related to the uncertainty in momentum by: where is the mass of the proton. Solving for :
Step 4: Conclusion.
Thus, the uncertainty in velocity is approximately 4.0 .

Step 1: Probability of finding the particle.
The probability of finding the particle between and is given by the integral of over that range:
Step 2: Square the wave function.
The wave function is given by: Thus:
Step 3: Set up the integral.
The probability is:
Step 4: Simplify the integral.
Use the identity to simplify the integral: This simplifies to: Evaluating the integral, we get:
Step 5: Conclusion.
Thus, the probability of finding the particle between and is 0.5.

Step 1: Energy levels of a particle in a box.
For a particle in a cubic box, the energy levels are quantized. The ground state energy is , and the energy levels are given by: where is a positive integer, is Planck's constant, is the particle's mass, and is the length of the box.
Step 2: Finding the energy of the first excited state.
The ground state energy corresponds to . The first excited state corresponds to the next value of . So, the energy of the first excited state is:
Step 3: Conclusion.
Thus, the energy of the first excited state is , so the correct answer is (C).

Step 1: Particle in infinite square well potential.
The energy eigenstates in an infinite square well are given by with corresponding energies Step 2: Superposed state.
The superposed state is Step 3: Expectation value of momentum.
Since the wavefunctions and are odd and even, respectively, the expectation value of momentum for a superposition of odd and even states is zero. Thus, Step 4: Uncertainty in momentum.
The uncertainty in momentum can be calculated as Step 5: Expectation value of energy.
The expectation value of energy is Step 6: Final Answer.
Hence, the correct answers are (A) and (B).