1. A particle is in an infinite square well of width L. Calculate the ground-state energy if (a) the particle is a proton and L = 0.1 nm, a typical size for a molecule (b) the particle is a proton and L = 1 fm, a typical size for a nucleus.

1. Louis de Broglie postulated all matter has both a particle and wave nature. The wavelength of any matter wave is given by λ = h/p, where p is the linear momentum p = mv and both p and v are vectors. a) World class marathon runners can run at an average pace of about 3.00 min/km. What is the wavelength, momentum and kinetic energy of a 55 kg runner racing at this average speed? b) What is the uncertainty in the runner's velocity if the position is to be determined within one wavelength?

1) An electron is confined to a square box of length L, and the walls of that box are infinitely high. The zero-point energy (ZPE) is defined as the minimal energy that corresponds to the smallest quantum number n. What would be the length of the box L such that the ZPE of the electron located inside this box is equal to its rest mass energy mec2?

7. One electron is trapped in a one-dimensional square well potential with infinitely high sides. a. If you have a probe that has a width for electron detection Ax = 0.00350L in the x direction, for the first excited state ( n =2), what is the probability that the electron is found in the probe when it is centered at x = L/4, (hint: you can use an approximation for this - you do not need to do an integral)? b. What is the average number of electrons that you would detect using the probe described in part "b." centered at x = L/4, ifthe electron is in the first excited state (n = 2) for each experiment and you repeat the experiment N, =100,000 times?

4. Use the variational principle to estimate the ground state energy of a particle in the potential (∞0 x < 0 U(x) = \cx x≥0 Take xe-bx as a trial function.

4. Normalize the following wavefunctions 4 55 (a) v(x) = sin (#2); =sin(); for a particle in a 1D box of length L. (b) (2) = xe-z|2 (c) (x) = e(x²/a²)+(ikz) 5. In a region of space, a particle with mass m and with zero energy has a time- independent wave-function (x) = Ae-2/12, where A and L are constants. Use your knowledge of the Schrödinger equation to determine the potential energy V(x) of the particle. Plot the potential function? What is the minimum potential energy for the particle, if it is an electron and L = 1 fm? Is this potential repulsive or attractive?