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Chapter 6: Q14E (page 224)

Verify that the reflection and transmission probabilities given in equation (6-7) add to 1.

Short Answer

Expert verified

The values of T and R in equations (6-7) add up to 1

Step by step solution

01

Concept involved

The ratio of amplitudes of transmitted and incident waves is called theTransmission coefficient.

Similarly, the ratio of amplitudes of reflected and incident waves is called theReflection coefficient

02

Formula used

The Transmission coefficient (T) and the Reflection coefficient (R) are given by the formulas:

T=4EE-U0E+E-U02R=E-E-U02E+E-U02

03

Calculation

T+R=4EE-U0E+E-U02+EE-U0E+E-U02T+R=E+E-U0-2EE-U0+4EE-U0E+E-U02T=RE2+E-U02+2EE-U0E+E-U02T+R=E+E-U02E+E-U02⇒T+R=1

So, it can be clearly seen that the values of T and R in equations (6-7) add up to 1

Font of the math type should be arial

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Most popular questions from this chapter

As we learned in example 4.2, in a Gaussian function of the formψ(x)αe-(x2/2ε2)is the standard deviation or uncertainty in position.The probability density for gaussian wave function would be proportional toψ(x)squared:e-(x2/2ε2). Comparing with the timedependentGaussian probability of equation (6-35), we see that the uncertainty in position of the time-evolving Gaussian wave function of a free particle is given by

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(a) Using the relativistically correct relationship among energy, momentum and mass, show that the correct dispersion relation is

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(b) Show that in the limit of low speed (small p and k) and ignoring mass/internal energy, this expression aggress with that of equation (6-23).

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