The 1D Particle in a Box

Questions

The Potential

The potential energy is 0 within an interval (box) of length L and infinity elsewhere. The latter constraint simply means that the particle must have a zero probability of being outside the interval.

For example, one choice of coordinates where the center of the 'box' is taken to be 0 on the x axis gives the potential energy function the form:

The Wavefunction

; n=1,2,3,......

where N is the normalization constant

Question 1

Find the normalization constant N for the ground (n=1) and first excited (n=2) states of the 1D particle in a box. Remember that the purpose of this constant to to ensure that the following condition holds. Namely, the probability function y(x)*y(x) integrated over all of the space the particle can roam must equal 1. That is, the particle must be somewhere in the box.

normalization condition:

Question 2

What is the ground state energy (a.k.a. the zero-point energy) of the 1-D Particle in a box? How does this zero point energy change with the size of the box and the mass of the particle? Also calculate the energy of the first excited state. Note that ground state energy is just the eigenvalue of the Hamiltonian operator applied to y1(x) and the energy of the first excited state is just the eigenvalue fo the Hamiltonian applied to y2(x).

Hint: Make sure that you have properly included the potential energy component of the Hamiltonian, V(x). This is really easy to do.

Question 3

What is the ultimate source of the zero point energy and energy quantization for the particle in a box?

Question 4

When an operator such as the Hamiltonian has more than one eigenfunction, like we have here with y1(x) and y2(x), these functions have a special relationship to one another. You can see what this relationship is if you calculate the overlap, S, of these two functions. Overlap is defined as

Calculate the overlap of y1(x) and y2(x) for the 1D PIB.

Question 5

a.  Calculate the expectation value of the position, x, for the ground state of the 1D PIB.

b.  Calculate the expectation value of the momentum, p, for the ground state of the 1D PIB

c.  Calculate the expectation value of the square of the position, x2, for the ground state of the 1D PIB

d.  Calculate the expectation value of the square of momentum, p2, for the ground state of the 1D PIB.

e.  Given that one way of determining the uncertainty of a variable such as Dx or Dp is

Question 6

Suppose you have a proton (H+) stuck between two positively charged cell membranes that are 1.0 nm apart (assume that electrical potential between the membranes is essentially zero while and jumps very quickly to a huge value, i.e. infinity, at the membrane itself).. What is zero point energy of the proton? What is the energy of the lowest excited state of the proton? What wavelength of light would be used to probe this excitation? What would the zero point energy of an electron caught between two similar negatively charged membranes? What about first excited state of such an electron?

Question 7

A scanning tunneling microscope works because electrons escape from the surface of a semiconducting surface by (you guessed it) tunnelling. Suppose that an electron is known to have a non-zero probability of tunnelling 10.0 nm into a potential barrier, how far would a muon under similar conditions tunnel into the barrier? (A muon is a negatively charged particle with a mass 100 times that of an electron).

Question 8

Write the wavefunction for a particle in a 2-D box of length L on both sides.

Question 9

What is the zero point energy of this wavefunction? What is the degeneracy of the 2D Particle in a box wavefunction at its zero point energy?

Question 10

Consider a roatation by 90 degrees of the ground state wavefunction (leaving the coordinate system unchanged). Have you changed the energy? Have you changed the 'appearance' (i.e. distribution of particle density) of the wavefunction? If the wavefunction is degenerate with another wavefunction and rotation does change the particle distribution, how does the new particle distribution compare to that of the 'degenerate' wavefunction?

Question 11

What is the energy and degeneracy of the 1st excited state of the 2D PIB (not including states degenerate to the one found in question 10. Consider a rotation by 90 degrees as in question 10. Have you changed the distribution of particle density of the wavefunction. Is there a degenerate partner to this state?

Question 12

What is the 3rd excited state energy and degeneracy?

Question 13

Consider a rectangular box of dimensions L1 x L2 (L1=1.5 L2). What is the zero point energy and degeneracy of the ground state wavefunction? What about the first excited state?

Question 14

Consider a particle confined to a cube of dimension LxLxL. Write the ground state wavefuction of this particle

Question 15

What is the zero point energy and degeneracy of this ground state. Do you see a pattern?