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A one-dimensional potential has the following form:
where V0 and b are positive constants.
Find V0 as a function of b such that there is just one bound state, of about zero binding energy, for a particle of mass M.
where V0 and b are positive constants.
Find V0 as a function of b such that there is just one bound state, of about zero binding energy, for a particle of mass M.
- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A one-dimensional potential has the following form:whereV0and bare pos...
To solve the bound state problem, we require that −V0<E<0. The relevant boundary conditions are ψ(0)=0, due to the infinitely high barrier at x=0 and limx→∞ψ(x)=0. The solution to the time-independent Schrödinger equation is:
where
Diving these two equations, we obtain,
Note that equation (1) implies that,
We can also rewrite equations (2) and (3) as,
We require
The requirement of k > 0 is conventional, since the sign of k can be absorbed into the definition of the constant A.
Equations (4) and (5) can be solved graphically by looking for intersection of the function
with the circle of radius 
in the first quadrant of the xy plane (where x=kb and y=kb ).
It is clear from the figure above that if V0 is less than some critical value, then the radius of the circle,
in which case there are no intersections in the first quadrant, and therefore no bound states. The critical value of of V0 is obtained by setting 
That is,
which yields,
where
Diving these two equations, we obtain,
Note that equation (1) implies that,
We can also rewrite equations (2) and (3) as,
We require
The requirement of k > 0 is conventional, since the sign of k can be absorbed into the definition of the constant A.Equations (4) and (5) can be solved graphically by looking for intersection of the function
with the circle of radius in the first quadrant of the xy plane (where x=kb and y=kb ).It is clear from the figure above that if V0 is less than some critical value, then the radius of the circle,
in which case there are no intersections in the first quadrant, and therefore no bound states. The critical value of of V0 is obtained by setting That is,
which yields,
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A one-dimensional potential has the following form:whereV0and bare positive constants.Find V0as a function of bsuch that there is just one bound state, of about zero binding energy, for a particle of mass M.a)b)c)d)Correct answer is option 'A'. Can you explain this answer?
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