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Chapter 5: 6CQ (page 186)

When is the temporal part of the wave function 0 ? Why is this important?

Short Answer

Expert verified

Answer:

The likelihood of discovering the probability would be 0 if the temporal part of a wave function was zero, implying that the particle would have vanished.

Step by step solution

01

Theory of wave function

The wave function ψitself has no physical significance. However, the amplitude of wave function corresponds ψto and the square of the wave functionψ relates to the photon density, the number of photons present in a region, it relates to electron density in a certain region.

02

Conclusion

Therefore, using the square of wave function, we can measure of the probability that the electron can be found within a particular tiny volume of the atom.

Hence, The likelihood of discovering the probability would be 0 if the temporal part of a wave function was zero, implying that the particle would have vanished.

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Most popular questions from this chapter

Consider the delta well potential energy:

U(x)={0x≠0-∞x=0

Although not completely realistic, this potential energy is often a convenient approximation to a verystrong, verynarrow attractive potential energy well. It has only one allowed bound-state wave function, and because the top of the well is defined as U = 0, the corresponding bound-state energy is negative. Call its value -E0.

(a) Applying the usual arguments and required continuity conditions (need it be smooth?), show that the wave function is given by

ψ(x)=(2mE0h2)1/4e-(2mE0/ℏ)|x|

(b) Sketch ψ(x)and U(x) on the same diagram. Does this wave function exhibit the expected behavior in the classically forbidden region?

A study of classical waves tells us that a standing wave can be expressed as a sum of two travelling waves. Quantum-Mechanical travelling waves, discussed in Chapter 4, is of the form ψ(x,t)=Aei(kx=ωt). Show that the infinite well’s standing wave function can be expressed as a sum of two traveling waves.

A finite potential energy function U(x) allows ψ(x) the solution of the time-independent Schrödinger equation. to penetrate the classically forbidden region. Without assuming any particular function for U(x) show that b(x) must have an inflection point at any value of x where it enters a classically forbidden region.

The quantized energy levels in the infinite well get further apart as n increases, but in the harmonic oscillator they are equally spaced.

  1. Explain the difference by considering the distance “between the walls” in each case and how it depends on the particles energy
  2. A very important bound system, the hydrogen atom, has energy levels that actually get closer together as n increases. How do you think the separation between the potential energy “walls” in this system varies relative to the other two? Explain.

The harmonic oscillator potential energy is proportional to x2, and the energy levels are equally spaced:

En∝(n+12). The energy levels in the infinite well become farther apart as energy increases: En∝n2.Because the functionlimb→∞|x/L|bis 0 for|x|<Land infinitely large for|x|>L. the infinite well potential energy may be thought of as proportional to |x|∞.

How would you expect energy levels to be spaced in a potential well that is (a) proportional to |x|1and (b) proportional to -|x|-1? For the harmonic oscillator and infinite well. the number of bound-state energies is infinite, and arbitrarily large bound-state energies are possible. Are these characteristics shared (c) by the |x|1well and (d) by the-|x|-1well? V
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