Problem 4: Variational principle and bounds (3 parts, 30 points)
PART A (15 points possible)
Consider a particle of mass m in a box of size L so the wave function vanishes at x=±L/2.
Find an upper bound E∗ on the ground state energy Egs using a simple trial wavefunction: a quadratic function of x which vanishes at ±L/2.
Useful integral: ∫10(1−u2)2du=815.
Enter 'hbar' for ℏ, m and L as needed.
Egs≤E∗= - unanswered
Compare the bound E∗ with the exact ground state energy E0:
E∗E0= - unanswered
PART B (5 points possible)
Which of the following trial wavefunctions could be used to find an upper bound on the energy of the first excited level?
(x−L/2)(x+L/2) <text> (x−L/2)(x+L/2)</text> - incorrect x(x−L/2)(x+L/2) x2 cos(πx/L) xx2−L2/4
You have used 1 of 1 submissions
PART C (10 points possible)
Consider a particle of mass m in a box of size L so the wave function vanishes at x=±L/2.
Use the uncertainty principle to calculate a lower bound on the ground state energy (Hint: assume the maximum possible value for Δx is Δx=L/2.)
Enter 'hbar' for ℏ, m and L as needed.
Egs≥