Homework 3

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Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q1

Starting with the ansatz wavefunction in Equation 3.1.18, derive the time-independent Schrödinger equation from the more general time-dependent Schrödinger equation.

Q2

Write the Schrödinger equation for a particle of mass \(m\) moving in a 2-dimensional space with the potential energy given by \(V(x, y) = -\dfrac {(x^2 + y^2)}{2}\).

Q3

Demonstrate that the function \(e^{ikx}\) is an eigenfunction of the momentum operator.

Q4

Consider a particle of mass \(m\) in a one-dimensional box of width \(L\). Calculate the general formula for energy of the transitions between neighboring states. How much energy is required to excite the particle from the \(n=1\) to \(n=2\) state?

Q5

Consider a particle of mass \(m\) in a two-dimensional square box with sides \(L\). Calculate the four lowest (different!) energies of the system. Write them down in the increasing order with their principal quantum numbers.

Q6

How will your answer to question 5 change if you consider a two-dimensional rectangular box with dimensions \(L_x\) and \(L_y\), where \(\dfrac{L_x}{L_y}=2\)? Write down the four lowest energies of the system in the increasing order with their principal quantum numbers.

Q7

Find the expectation value of \(x\) for n=1 in a 1-d particle in the box with length \(L\), and particle mass \(m\).

Q8

Find the expectation value of \(x^2 \) for n=1 in a 1-d particle in the box with length \(L\), and particle mass \(m\).

Q9

Find the expectation value of \(p_x\) for n=1 in a 1-d particle in the box with length \(L\), and particle mass \(m\).

Q10

Find the expectation value of \(p_x^2\) for n=1 in a 1-d particle in the box with length \(L\), and particle mass \(m\).

Using \(\Delta{q^2} = \langle q^2 \rangle - \langle q \rangle^2 \) and the expectation values you have now calculated, prove the Heisenberg Uncertainty principle holds true for the n=1 level of of the particle in the box.

Q11

What is the expectation value of position for a particle in a box of length (\(L\)) in the ground eigenstate (n=1)? What about for the first excited eignestate (n=2). Explain the difference.

Q12

What is the most probably position for a particle in a box of length (\(L\)) in the ground eigenstate (n=1)? What about for the first excited eignestate (n=2). Explain the difference.

Q13

The uncertainty of a position of a particle (\(\Delta x\)) is defined as

\[ \Delta x = \sqrt{ \langle x^2 \rangle - \langle x \rangle ^2} \]

For the particle in the box at the ground eigenstate (\(n=1\)), what is the uncertainty in the value \( x \)?

Q14

Why should wavefunctions be normalized? How is the expectation formula changed if unnormalized wavefunctions were used and why?

\[ \langle x \rangle = \int_{-\infty}^{+\infty} \psi^*(x) x \psi(x) dx \]