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Chapter 5: Q31E (page 188)

Verify that solution (5-19) satisfies the Schrodinger equation in form (5.18).

Short Answer

Expert verified

The solution of the Schrodinger equation is

d2ψ(x)dx2=Aα2exp±αxd2ψ(x)dx2=α2ψ(x)

Step by step solution

01

Time-dependent equation of Schrodinger.

Time-dependent equation of Schrodinger

d2ψ(x)dx=2m(Uo(x)-E)h2ψ(x)………………(1)

and the wave function is provided

ψ(x)=Aexp±αx,α=2m(Uo(x)-E)h2………………(2)

02

Schrodinger equation

Differentiate the wave function twice and replace into the Schrodinger equation to ensure that it satisfies the Schrodinger equation (1).

dψ(x)dx=±Aαexp±αx=±αψ(x)d2ψ(x)dx2=Aα2exp±αx=α2ψ(x)

So, the Schrodinger equation isd2ψ(x)dx2=Aα2exp±αxd2ψ(x)dx2=α2ψ(x).

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Most popular questions from this chapter

Calculate the uncertainty in the particle’s position.

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.

Using equation (23), find the energy of a particle confined to a finite well whose walls are half the height of the ground-state infinite well energy, . (A calculator or computer able to solve equations numerically may be used, but this happens to be a case where an exact answer can be deduced without too much trouble.)

An electron is trapped in a quantum well (practically infinite). If the lowest-energy transition is to produce a photon ofwavelength. Whatshould be the well’s width?

A 2kg block oscillates with an amplitude of 10cm on a spring of force constant 120 N/m .

(a) In which quantum state is the block?

(b) The block has a slight electric charge and drops to a lower energy level by generating a photon. What is the minimum energy decrease possible, and what would be the corresponding fractional change in energy?
See all solutions

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