The probability of finding a particle in a unit volume at position x is equal to

Question

The probability of finding a particle in a unit volume at position x is equal to the square of the wave function psi (x). which is a solution to the Schrodinger Equation: d^2 psi/dx^2 = 2m/h^2 (U -E) psi In words: At position x, the curvature of the wavefunction (d^2 psi/dx^2) is proportional to (U - E) psi. The sign of (U - E) dictates whether the solution is sinusoidal (what sign?) or exponential (what sign?). Total energy E is a constant, determined by the solution of the wave equation. For a particle trapped in a potential well given by U(x), the equation is satisfied only for discrete energies, E_n, in order to meet the specific boundary conditions. Consider a particle trapped in the following potential well. What is n for this solution to the wave equation? (The ground state has n = 1.) n = _____. Notice that at the boundaries, the wave function psi and its slope d psi/dx are continuous. U - E is positive in Regions ____. psi curves away, from the x-axis. Classically forbidden region. U - E is negative in Regions ______. psi curves toward the x-axis. Classically allowed region.

Explanation / Answer

3) If (U-E) is +ve the solution is exponential , if it is -ve then the solution is sinusoidal

d2/dx2 (sinfunc) = -const*(sinfunc) where as d2/dx2 (exponet func) = +const*(exponet func)

4) The given wave function is sinusoidal one and for each quantum number the wave makes one half-cycle in the well with penetration into the wall of the well.

The given wave has 5 half-cycles in region C and 2 half-cycles in region B

hence n= 7

(U-E) >0 in regions A and D

it is -ve in regions B and C

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