- #1
mrguru34
- 11
- 0
I have a problem:
Let V0(x) denote the potential corresponding to the in nite
square well (`box') extending from x = 0 to x = a . Let
us replace it by the potential
V (x) = 0 ; |x| [tex]\leq[/tex] a/2
[tex]\infty[/tex]; |x| [tex]\geq[/tex] a/2
i want to fi nd the solutions of the Schrodinger equation associated
with V using the knowledge of the system described by V0.
(i)What is the mathematical relationship between V0(x) and V (x)?
Sketch both to illustrate this relationship.
(ii) Write down the time-independent Schrodinger equation for a particle
of mass m moving in the potential V (x) described by wave
functions [tex]\Phi[/tex](x).
(iii) Let [tex]\Psi[/tex](x) denote the solutions of the time-independent Schrodinger
equation with potential V0(x) derived in class. Using (i) express
the `new' wave functions [tex]\Phi[/tex](x) in terms of the `old' ones, [tex]\Psi[/tex](x).
(iv) Write down explicit expressions for the wave functions [tex]\Psi[/tex](x).
[Hint: distinguish between n even and n odd.]
(v) Show that the wave functions [tex]\Phi[/tex] obey the appropriate boundary
conditions for motion in the potential V (x).
(vi) Write down the energy eigenvalues En associated with the eigenfunctions
[tex]\Phi[/tex] (no calculation required).
Thanks for your help.
It would be much appreciated if someone could actually explain very simply what's going on here aswell
Thanks again
Let V0(x) denote the potential corresponding to the in nite
square well (`box') extending from x = 0 to x = a . Let
us replace it by the potential
V (x) = 0 ; |x| [tex]\leq[/tex] a/2
[tex]\infty[/tex]; |x| [tex]\geq[/tex] a/2
i want to fi nd the solutions of the Schrodinger equation associated
with V using the knowledge of the system described by V0.
(i)What is the mathematical relationship between V0(x) and V (x)?
Sketch both to illustrate this relationship.
(ii) Write down the time-independent Schrodinger equation for a particle
of mass m moving in the potential V (x) described by wave
functions [tex]\Phi[/tex](x).
(iii) Let [tex]\Psi[/tex](x) denote the solutions of the time-independent Schrodinger
equation with potential V0(x) derived in class. Using (i) express
the `new' wave functions [tex]\Phi[/tex](x) in terms of the `old' ones, [tex]\Psi[/tex](x).
(iv) Write down explicit expressions for the wave functions [tex]\Psi[/tex](x).
[Hint: distinguish between n even and n odd.]
(v) Show that the wave functions [tex]\Phi[/tex] obey the appropriate boundary
conditions for motion in the potential V (x).
(vi) Write down the energy eigenvalues En associated with the eigenfunctions
[tex]\Phi[/tex] (no calculation required).
Thanks for your help.
It would be much appreciated if someone could actually explain very simply what's going on here aswell
Thanks again