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Chapter 6: Q3CQ (page 223)

Why is the topic of normalization practically absent from Sections 6.1 and 6.2?

Short Answer

Expert verified

We don’t normalize with multiple particles as with a single particle.

Step by step solution

01

Definition of normalization

Normalization is defined as the scaling of all wave functions of a quantum state to an exact form such that all probabilities of finding a particle anywhere in space add up to 1.

02

Explanation and conclusion

In this section, plane waves are used for the representation of a single particle or a group of particles behaving as one coherent wave and studying its reflection and transmission coefficients. For multiple particles, we usually don’t normalize for unit probability but seek the multiplicative coefficients.

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Most popular questions from this chapter

In the E>Uopotential barrier, there should be no reflection when the incident wave is at one of the transmission resonances. Prove this by assuming that a beam of particles is incident at the first transmission resonance, E=Uo+(π2h2/2mL2), and combining continuity equations to show thatB=0. (Note: k’ is particularly simple in this special case, which should streamline your work.)

Exercise 39 gives a condition for resonant tunneling through two barriers separated by a space width of2s, expressed I terms of factorβgiven in exercise 30. Show that in the limit in which barrier widthL→∞, this condition becomes exactly energy quantization condition (5.22) for finite well. Thus, resonant tunneling occurs at the quantized energies of intervening well.

Particles of energy Eare incident from the left, where U(x)=0, and at the origin encounter an abrupt drop in potential energy, whose depth is -3E.

  1. Classically, what would the particles do, and what would happen to their kinetic energy?
  2. Apply quantum mechanics, assuming an incident wave of the formψinc=eikx, where the normalization constant has been given a simple value of 1, determine completely the wave function everywhere, including numeric values for multiplicative constants.
  3. What is the probability that incident particles will be reflected?

How should you answer someone who asks, “In tunneling through a simple barrier, which way are particles moving, in the three regions--before, inside, and after the barrier?”

A beam of particles of energy E incident upon a potential step ofU0=(5/4)E is described by wave function:ψinc(x)=eikx

  1. Determine the reflected wave and wave inside the step by enforcing the required continuity conditions to obtain their (possibly complex) amplitudes.
  2. Verify the explicit calculation the ratio of reflected probability density to the incident probability density is 1.
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