A particle is confined to a 1-D box of length L. Deduce the location (in terms of L) of the positions within the box at which the particle is most likely to be found when the quantum number of the particles is (a) n =1, (b) n=2, and (c) n =3.

If the energy of the first excited state of the electron in the box is 25.0 eV, what is the width of the box?

If the uncertainty in the y -component of a proton's position is 2.0 pm, find the minimum uncertainty in the simultaneous measurement of the proton's y -component of velocity. What is the minimum uncertainty in the simultaneous measurement of the proton's x -component of velocity?

A particle with mass m moving along the x-axis and its quantum state is represented by the following wave function: (x,t)={0,x0Ax e ax e iEt/h,x0, where a a=2.01010m1. (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval 0xL. (c) Find the expectation value of position. (d) Find the expectation value of kinetic energy.

Is it possible that when we measure the energy of a quantum particle in a box, the measurement may return a smaller value than the ground state energy? What is the highest value of the energy that we can measure for this particle?

(a) Using the Pauli exclusion principle and the rules relating the allowed values of the quantum numbers (n,l,ml,ms), prove that the maximum number of electrons in a subshell is 2n2. (b) In a similar manner, prove that the maximum number of electrons in a shell is 2n2.

Assume that a proton in a nucleus can be treated as if it were confined to a one-dimensional of width 10.0 fm. (a) What are the energies of the proton when it is in the states corresponding to n=1,n=2, and n=3? (b) What are the energies of the photons emitted when the proton makes the transitions from the first and second excited states to the ground state?

(a) If the position of a chlorine ion in a membrane is measured to an accuracy of 1.00 m, what is its minimum uncertainty in velocity, given its mass is 5.8601026 kg? (b) If the ion has this velocity, what is its kinetic energy in eV, and how does this compare with typical molecular binding energies?

Show that Planck’s constant has the dimensions of angular momentum.