10.6 Substitute the ground state into Schrödinger’s equation and determine the energy of this
state.
10.7 If the length Lis 1.0 Å, determine the probability of finding the particle in the ground
state between x=L/4 and x= 3 L/4.
10.8 If the length Lis 1.0 Å, calculate the longest wavelength transition for an electron that is
initially in the lowest-energy state.
10.9 If the length Lis 1.0 Å, calculate the wavelength of a photon that is absorbed after an elec-
tronmakes a transition from the ground state to the fourth excited state.
10.10 Assume there are six electrons in a box with a length of 10 nm: (a) What is the highest
occupied state? (b) What is the wavelength of the highest occupied state? (c) What is the
lowest unoccupied state? (d) What is the wavelength of the lowest unoccupied state?
(e) What is wavelength of the longest wavelength transition?
10.11 For the following questions assume that the potential is given by:
V(x≤−a) =∞
V(x≥+ 3 a) =∞
V(−a≤x≤+ 3 a) = 0(a) Write the boundary conditions for the problem.
(b) Write Schrödinger’s equation for this problem.
(c) Write the general solution to this problem.
(d) In terms of a, what is the wavelength of the ground state?10.12 For the following questions, assume that a particle is trapped in a box such that:
V (x<b) =∞
V (x ≥band x≤ 3 b) =V 0
V (x> 3 b) =∞where V 0 is a constant greater than zero.
(a) Write Schrödinger’s equation for this potential.
(b) What boundary conditions can be placed on the wavefunctions?
(c) What are the wavefunctions for this problem?
(d) What is the wavelength of the ground state?
(e) What is the energy of the ground state?10.13Assume that a two-dimensional box is rectangular with a length of 10 Å along xand 20 Å
along y: (a) write Schrödinger’s equation for this problem; (b) write the ground-state wave-
function; (c) calculate the ground-state energy for a single electron; and (d) calculate the
longest wavelength transition when there is one electron in the box.
10.14 For the following two-dimensional problem assume that the potential is given by:
V(x≤− 2 a) =∞
V(x≥+ 3 a) =∞
V(−a≤x≤+ 3 a) = 0
V(y≤−b) =∞
V(y≥+ 2 b) =∞
V(−b≤y≤+ 2 b) = 0