(a)
The elements of M matrix in terms of elements of S matrix,
(a)
Answer to Problem 2.54P
The elements of M matrix in terms of elements of S matrix is
Explanation of Solution
The element B can be expressed as follows,
The element F can be expressed as follows,
Use the value of G from the equation (I) in (II).
From the equation (I) and (III), the matric M can be written as follows.
Similarly define the element G and solve for
Similarly define the element F and solve for
Use the value of B from the equation (V) in (VI).
From equation (V), and (VII).
From the equation (V),
From the equation (VII),
From the equation (V),
From the equation (VII),
Conclusion:
Therefore, the elements of M matrix in terms of elements of S matrix is
(b)
Show that M matrix for the combination is the product of the two M matrix for each section separately.
(b)
Answer to Problem 2.54P
Showed that M matrix for the combination is the product of the two M matrix for each section separately.
Explanation of Solution
Consider the Figure below.
Consider the matrix.
Consider the matrix.
Use equation (XIV) in (XIII).
Hence proved.
Conclusion:
Therefore, the M matrix for the combination is the product of the two M matrix for each section separately.
(c)
The matrix for the scattering from a single delta function potential at point a.
(c)
Answer to Problem 2.54P
The matrix for the scattering from a single delta function potential at point a is
Explanation of Solution
The wave function for the scattering from a single delta function potential at point a is
Using the continuity condition of the wave function, equation (XV) can be written as.
Using the discontinuity condition of the derivative of wave function, equation (XV) can be written as.
Multiply the equation (XVI) with
Multiply the equation (XVII) with
Add equation (XVIII) and (XIX), and solve for
Where,
Subtract equation (XIX) from (XVIII), and solve for
From equation (XX), and (XXI), the matrix can be written as follows.
Conclusion:
Therefore, the matrix for the scattering from a single delta function potential at point a is
(d)
The matrix for scattering from the double delta function, and the transmission coefficient for this potential.
(d)
Answer to Problem 2.54P
The matrix for scattering from the double delta function is
Explanation of Solution
The matrix
From the result of part (b), the matrix for scattering from the double delta function can be expressed as.
The transmission coefficient can be calculated using the equation (XXIV).
Simplify the equation (XXV).
Conclusion:
Therefore, The matrix for scattering from the double delta function is
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Chapter 2 Solutions
Introduction To Quantum Mechanics
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