Conditions for smoothness at regional boundaries for energy E less than barrier potential U 0 and verification that reflection coefficient B is expressed as A ( s i n h ( α L ) ( s i n h 2 ( α L ) + 4 α 2 k 2 / ( k 2 + α 2 ) 2 ) 1 2 ) e − i β .

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Answer 1

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Chapter 6, Problem 30E

(a)

To determine

Conditions for smoothness at regional boundaries for energy E less than barrier potential U0 and verification that reflection coefficient B is expressed as A(sinh( αL) ( sin h 2 ( αL )+ 4 α 2 k 2 / ( k 2 + α 2 ) 2 ) 1 2 )e−iβ .

(b)

To determine

Verification that reflection probability R in equation (6.16) is followed from this result.

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Suppose we know exactly two arbitrary distributions p(x|ωi) and priors P(ωi) in a d-dimensional feature space. (a) Prove that the true error cannot decrease if we first project the distributions to a lower dimensional space and then classify them. (b) Despite this fact, suggest why in an actual pattern recognition application we might not want to include an arbitrarily high number of feature dimensions.

Legrende polynomials The amplitude of a stray wave is defined by: SO) =x (21+ 1) exp li8,] sen 8, P(cos 8). INO Here e is the scattering angle, / is the angular momentum and 6, is the phase shift produced by the central potential that performs the scattering. The total cross section is: Show that: 'É4+ 1)sen² 8, .

Consider the "step" potential: V(x) = (a) Calculate the reflection coefficient, for the case E 0. (b) Calculate the reflection coefficient for the case E > Vo. (c) For a potential such as this, which does not go back to zero to the right of the barrier, the transmission coefficient is not simply |F12/A2 (with A the -Vo AV(x) Scattering from a "cliff" incident amplitude and F the transmitted amplitude), because the transmitted wave travels at a different speed. Show that T = E-Vo F1² E |A|² X for E> Vo. Hint: You can figure it out using Equation gantly, but less informatively-from the probability current ( What is T, for E Vo, calculate the transmission coefficient for the step potential, and check that T + R = 1.

Answer 2

Question

Chapter 6, Problem 30E

(a)

To determine

Conditions for smoothness at regional boundaries for energy E less than barrier potential U0 and verification that reflection coefficient B is expressed as A(sinh( αL) ( sin h 2 ( αL )+ 4 α 2 k 2 / ( k 2 + α 2 ) 2 ) 1 2 )e−iβ .

(b)

To determine

Verification that reflection probability R in equation (6.16) is followed from this result.

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Answer 3

Question

Chapter 6, Problem 30E

(a)

To determine

Conditions for smoothness at regional boundaries for energy E less than barrier potential U0 and verification that reflection coefficient B is expressed as A(sinh( αL) ( sin h 2 ( αL )+ 4 α 2 k 2 / ( k 2 + α 2 ) 2 ) 1 2 )e−iβ .

(b)

To determine

Verification that reflection probability R in equation (6.16) is followed from this result.

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Answer 4

Question

Chapter 6, Problem 30E

(a)

To determine

Conditions for smoothness at regional boundaries for energy E less than barrier potential U0 and verification that reflection coefficient B is expressed as A(sinh( αL) ( sin h 2 ( αL )+ 4 α 2 k 2 / ( k 2 + α 2 ) 2 ) 1 2 )e−iβ .

(b)

To determine

Verification that reflection probability R in equation (6.16) is followed from this result.

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Answer 5

Chapter 6, Problem 30E

(a)

To determine

Conditions for smoothness at regional boundaries for energy E less than barrier potential U0 and verification that reflection coefficient B is expressed as A(sinh( αL) ( sin h 2 ( αL )+ 4 α 2 k 2 / ( k 2 + α 2 ) 2 ) 1 2 )e−iβ .

(b)

To determine

Verification that reflection probability R in equation (6.16) is followed from this result.

Answer 6

Chapter 6, Problem 30E

(a)

To determine

Conditions for smoothness at regional boundaries for energy E less than barrier potential U0 and verification that reflection coefficient B is expressed as A(sinh( αL) ( sin h 2 ( αL )+ 4 α 2 k 2 / ( k 2 + α 2 ) 2 ) 1 2 )e−iβ .

Answer 7

Chapter 6, Problem 30E

(a)

To determine

Conditions for smoothness at regional boundaries for energy E less than barrier potential U0 and verification that reflection coefficient B is expressed as A(sinh( αL) ( sin h 2 ( αL )+ 4 α 2 k 2 / ( k 2 + α 2 ) 2 ) 1 2 )e−iβ .

Answer 8

(b)

To determine

Verification that reflection probability R in equation (6.16) is followed from this result.

Answer 9

(b)

To determine

Verification that reflection probability R in equation (6.16) is followed from this result.

Answer 10

Expert Solution & Answer

Answer 11

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Answer 12

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Answer 13

Ch. 6 - Prob. 1CQ Ch. 6 - Prob. 2CQ Ch. 6 - Prob. 3CQ Ch. 6 - Prob. 4CQ Ch. 6 - Prob. 5CQ Ch. 6 - Prob. 6CQ Ch. 6 - Prob. 7CQ Ch. 6 - Prob. 8CQ Ch. 6 - Prob. 9CQ Ch. 6 - Prob. 10CQ

Conditions for smoothness at regional boundaries for energy E less than barrier potential U 0 and verification that reflection coefficient B is expressed as A ( s i n h ( α L ) ( s i n h 2 ( α L ) + 4 α 2 k 2 / ( k 2 + α 2 ) 2 ) 1 2 ) e − i β .

Solution Summary: The author explains the Schrödinger wave function and its spatial derivatives are continuous across the boundaries between three different regions. Smoothness conditions and reflection coefficient are determined from the principle of continuity.

Conditions for smoothness at regional boundaries for energy E less than barrier potential U0 and verification that reflection coefficient B is expressed as A(sinh( αL) ( sin h 2 ( αL )+ 4 α 2 k 2 / ( k 2 + α 2 ) 2 ) 1 2 )e−iβ .

Verification that reflection probability R in equation (6.16) is followed from this result.

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Chapter 6 Solutions