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noblegas
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Homework Statement
A particle that moves in three dimensions is trapped in a deep spherically symmetric potential V(r):
V(r)=0 at r<r_0
infinity at r>= r_0
where r_0 is a positive constant. The ground state wave function is spherically symmetric , so the radial wave function u(r) satistifies the 1-dimensional schrodinger energy eigenvalue equation (-[tex]\hbar[/tex]2/2m)*d^2/dr^2*u(r)+V(r)*u(r)=Eu(r) with the boundary condition u(0)=0 (eqn u(r)=r*phi(r)
a) Explain why, in the potential well in the equation V(r)=0 at r<r_0
infinity at r>= r_0, the wave function is forced to vanish at r=r_0
b) Using the known boundary conditions on the radial wave function u(r) at r=0 and r=r_0 , find the ground state energy of the particle of this potential well
Homework Equations
(-[tex]\hbar[/tex]2/2m)*d^2/dr^2*u(r)+V(r)*u(r)=Eu(r)
u(r)=r*phi(r)
The Attempt at a Solution
a)I don't exactly know what they mean by vanish in this context. Does V vanish when V=0? I know in a classically allowed region, when a particle just sits in a well and there is no external energy being transferred to the particle, then V=0.
b) To begin this problem, I should probably plug u(r) into (-[tex]\hbar[/tex]2/2m)*d^2/dr^2*u(r)+V(r)*u(r)=Eu(r). Not sure how following this procedure will help me find the value for the ground state energy.