1. Consider a potential V(x) = V(-x) with two minima at x = ±a. (a) Explain why the energy eigenfunctions can be chosen to be wavefunctions of definite parity under x-x, that is the energy eigenfunctions can be chosen to obey either E(x) = VE(-x) (even parity) or VE(x) = -VE(-x) (odd parity). (b) As a simple example of a double well potential consider the potential V(x) = Ah² 2ma 6(x-a)+(x+a) with >0. Derive an equation which determines the energy eigenvalue for the lowest energy solution with even parity. (c) The equation you found cannot be solved in closed form, but the number of positive parity energy eigenvalues can be determined by plotting different terms in the equation and using graphical means to determine the number of solutions. How many solutions do you find and how does the number of solutions depend on A? 2. Consider scattering in the delta function potential we introduced in class, V(x) = - ħ2 2mL(x) Suppose we have a solution to the Schrödinger equation with E> 0 which has the form (x) = A₁eikx + A₁e-ikx, x < 0 ikx x > 0. (x) = Aze + Aze-ikx, Physically this corresponds to having both incoming and outgoing fluxes on both sides of the potential, rather than having just an incoming flux from the left as for the example analyzed in class. Show that = S A₁ (M)-(X) where S is a two by two matrix, and find the matrix S. This matrix is called the S-matrix or scattering matrix in more advanced treatments of scattering. Note that the S-matrix relates the coefficients of "outgoing" waves to those of "incoming" waves. Verify that S is a unitary matrix. This is a reflection of conservation of probability: what comes in must go out. Note: Although we analyzed scattering with this potential using Green functions in class, you can solve this problem by imposing the correct boundary conditions on the wave function and its derivatives at x = 0.
1. Consider a potential V(x) = V(-x) with two minima at x = ±a. (a) Explain why the energy eigenfunctions can be chosen to be wavefunctions of definite parity under x-x, that is the energy eigenfunctions can be chosen to obey either E(x) = VE(-x) (even parity) or VE(x) = -VE(-x) (odd parity). (b) As a simple example of a double well potential consider the potential V(x) = Ah² 2ma 6(x-a)+(x+a) with >0. Derive an equation which determines the energy eigenvalue for the lowest energy solution with even parity. (c) The equation you found cannot be solved in closed form, but the number of positive parity energy eigenvalues can be determined by plotting different terms in the equation and using graphical means to determine the number of solutions. How many solutions do you find and how does the number of solutions depend on A? 2. Consider scattering in the delta function potential we introduced in class, V(x) = - ħ2 2mL(x) Suppose we have a solution to the Schrödinger equation with E> 0 which has the form (x) = A₁eikx + A₁e-ikx, x < 0 ikx x > 0. (x) = Aze + Aze-ikx, Physically this corresponds to having both incoming and outgoing fluxes on both sides of the potential, rather than having just an incoming flux from the left as for the example analyzed in class. Show that = S A₁ (M)-(X) where S is a two by two matrix, and find the matrix S. This matrix is called the S-matrix or scattering matrix in more advanced treatments of scattering. Note that the S-matrix relates the coefficients of "outgoing" waves to those of "incoming" waves. Verify that S is a unitary matrix. This is a reflection of conservation of probability: what comes in must go out. Note: Although we analyzed scattering with this potential using Green functions in class, you can solve this problem by imposing the correct boundary conditions on the wave function and its derivatives at x = 0.
Transcribed Image Text:1. Consider a potential V(x) = V(-x) with two minima at x = ±a.
(a) Explain why the energy eigenfunctions can be chosen to be wavefunctions of
definite parity under x-x, that is the energy eigenfunctions can be chosen to
obey either E(x) = VE(-x) (even parity) or VE(x) = -VE(-x) (odd parity).
(b) As a simple example of a double well potential consider the potential
V(x) =
Ah²
2ma
6(x-a)+(x+a)
with >0. Derive an equation which determines the energy eigenvalue for the
lowest energy solution with even parity.
(c) The equation you found cannot be solved in closed form, but the number of
positive parity energy eigenvalues can be determined by plotting different terms
in the equation and using graphical means to determine the number of solutions.
How many solutions do you find and how does the number of solutions depend
on A?
2. Consider scattering in the delta function potential we introduced in class,
V(x) = -
ħ2
2mL(x)
Suppose we have a solution to the Schrödinger equation with E> 0 which has the
form
(x) = A₁eikx
+ A₁e-ikx,
x < 0
ikx
x > 0.
(x) = Aze + Aze-ikx,
Physically this corresponds to having both incoming and outgoing fluxes on both
sides of the potential, rather than having just an incoming flux from the left as for
the example analyzed in class. Show that
= S
A₁
(M)-(X)
where S is a two by two matrix, and find the matrix S. This matrix is called
the S-matrix or scattering matrix in more advanced treatments of scattering. Note
that the S-matrix relates the coefficients of "outgoing" waves to those of "incoming"
waves. Verify that S is a unitary matrix. This is a reflection of conservation of
probability: what comes in must go out. Note: Although we analyzed scattering
with this potential using Green functions in class, you can solve this problem by
imposing the correct boundary conditions on the wave function and its derivatives
at x = 0.
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Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning