4. Potential well - Schrödinger Equation. Please help.
Posted: Fri Apr 29, 2022 11:27 am
4. Potential well - Schrödinger Equation.
Please help.
Problem 4) Potential Well. We consider a particle inside a potential well, corresponding to the potential energy <<0 U(x) = { 0 U. 0<x<L, 0 >L (2) where U, >0. a) Solve the time independent Schrödinger equation for each of the tree regions x < 0 (region 1), 0 < x < L (region 2), and x > L (region 3) for a particle of mass m and with energy E<0. Write down the general solution for each of the three region b) Use the requirement that the integral over x must converge, to simply the solution in region 1 and 3 through discarding those terms which diverge for x + Foo in the regarding regions. After this procedure you should have one free parameter for region 1 and 3 and two free parameters for region 2 in your solution c) Impose continuity of the solution at x = 0 and x = L and use the resulting equations to express the normalization of the solution in region 1 and 3 in terms of the normalization in region 2, as well as the other paramter of the problem (mass, energy, U, etc.) d) Impose continuity of the first derivative at x = 0 and x = L. Use one of the resulting 2 equations to express one of the parameters of region 2 in terms of the other one (and param- eters of the solution). c) Use the remaining relation to obtain the relation ktan -() KL 2 = a, a= 2mE h V2m(U-ED) k= ħ (3) Hint 1: Follow closely the presentation discussed in class Hint 2: Possibly useful relation i. 1-eix 1+eis tan(3/2) =
- 4 Potential Well Schrodinger Equation Please Help 1 (182.95 KiB) Viewed 41 times
Problem 4) Potential Well. We consider a particle inside a potential well, corresponding to the potential energy <<0 U(x) = { 0 U. 0<x<L, 0 >L (2) where U, >0. a) Solve the time independent Schrödinger equation for each of the tree regions x < 0 (region 1), 0 < x < L (region 2), and x > L (region 3) for a particle of mass m and with energy E<0. Write down the general solution for each of the three region b) Use the requirement that the integral over x must converge, to simply the solution in region 1 and 3 through discarding those terms which diverge for x + Foo in the regarding regions. After this procedure you should have one free parameter for region 1 and 3 and two free parameters for region 2 in your solution c) Impose continuity of the solution at x = 0 and x = L and use the resulting equations to express the normalization of the solution in region 1 and 3 in terms of the normalization in region 2, as well as the other paramter of the problem (mass, energy, U, etc.) d) Impose continuity of the first derivative at x = 0 and x = L. Use one of the resulting 2 equations to express one of the parameters of region 2 in terms of the other one (and param- eters of the solution). c) Use the remaining relation to obtain the relation ktan -() KL 2 = a, a= 2mE h V2m(U-ED) k= ħ (3) Hint 1: Follow closely the presentation discussed in class Hint 2: Possibly useful relation i. 1-eix 1+eis tan(3/2) =