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Chapter 5: Q33E (page 188)

Verify that Asin(kx) + Bcos(kx)is a solution of equation (5 - 12).

Short Answer

Expert verified

The wave function equation satisfies the Schrodinger Equation.

Step by step solution

01

Wave function equation and Schrodinger Equation.

Schrodinger Equation,

d2ψdx2= -k2ψ(x),k =2mEh2...................1

Wave function equation,

ψ== A sin(kx)+B cos(kx)........1

02

Substituting (2) in to (1).

Substitute the values

d2ψ(x)dx2= -k2A sin(kx) -k2B cos(kx)dψ(x)dx= k A cos(kx) - k B sin(kx)d2ψ(x)dx2= -k2A sin(kx) -k2=k2ψxSo, thatwave function equation satisfies the Schrodinger Equation.

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Most popular questions from this chapter

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.

There are mathematical solutions to the Schrödinger equation for the finite well for any energy, and in fact. They can be made smooth everywhere. Guided by A Closer Look: Solving the Finite Well. Show this as follows:

(a) Don't throw out any mathematical solutions. That is in region Il (x<0), assume that (Ce+ax+De-ax), and in region III (x>L), assume thatψ(x)=Fe+ax+Ge-ax. Write the smoothness conditions.

(b) In Section 5.6. the smoothness conditions were combined to eliminate A,Band Gin favor of C. In the remaining equation. Ccanceled. leaving an equation involving only kand α, solvable for only certain values of E. Why can't this be done here?

(c) Our solution is smooth. What is still wrong with it physically?

(d) Show that

localid="1660137122940" D=12(B-kαA)andF=12e-αL[(A-Bkα)sin(kL)+(Akα+B)cos(kL)]

and that setting these offending coefficients to 0 reproduces quantization condition (5-22).

What is the product ofΔxandΔp(obtained in Exercise 83 and 85)? How does it compare with the minimum theoretically possible? Explain.

To describe the matter wave, does the function Asin(kx)cosωthave well-defined energy? Explain

Quantization is an important characteristic of systems in which a particle is bound in a small region. Why "small," and why "bound"?
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