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Chapter 5: Q55E (page 190)

To determine the classical expectation value of the position of a particle in a box is L2 , the expectation value of the square of the position of a particle in a box isrole="math" localid="1658324625272" L23 , and the uncertainty in the position of a particle in a box isL12 .

Short Answer

Expert verified

The uncertainty in the position of a particle in a box of lengthLisΔx=−x2−x¯2=L23−L24=L12.

Step by step solution

01

Step 1:

Given information: The classical probability per unit length of finding a particle in a box of lengthL is1L along the entire length of the box i.e.dPdx=1L .

02

Step 2:

The classical expectation value of the position of a particle in a box of length Lis

x¯=∫0LxdP=∫0LxdPdxdx=1L∫0Lxdx=1L×L22=L2.

The expectation value of the square of the position of a particle in a box of length role="math" localid="1658325211080" Lis

x2¯=∫0Lx2dP=∫0Lx2dPdxdx=1L∫0Lx2dx=1L×L33=L23

The uncertainty in the position of a particle in a box of length Lis

Δx=−x2−x¯2=L23−L24=L12

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Most popular questions from this chapter

Outline a procedure for predicting how the quantum-mechanically allowed energies for a harmonic oscillator should depend on a quantum number. In essence, allowed kinetic energies are the particle-in-a box energies, except the length Lis replaced by the distance between classical tuning points. Expressed in terms of E. Apply this procedure to a potential energy of the form U(x) = - b/x where b is a constant. Assume that at the origin there is an infinitely high wall, making it one turning point, and determine the other numing point in terms of E. For the average potential energy, use its value at half way between the tuning points. Again in terms of E. Find and expression for the allowed energies in terms of m, b, and n. (Although three dimensional, the hydrogen atom potential energy is of this form. and the allowed energy levels depend on a quantum number exactly as this simple model predicts.)

The potential energy shared by two atoms in a diatomic molecule, depicted in Figure 17, is often approximated by the fairly simple function U(x)=(ax12)-(bx6)where constants a and b depend on the atoms involved. In Section 7, it is said that near its minimum value, it can be approximated by an even simpler function—it should “look like” a parabola. (a) In terms ofa and b, find the minimum potential energy U (x0) and the separation x0 at which it occurs. (b) The parabolic approximation UP(x)=U(xo)+12κ(x-xo)2has the same minimum value at x0 and the same first derivative there (i.e., 0). Its second derivative is k , the spring constant of this Hooke’s law potential energy. In terms of a and b, what is the spring constant of U (x)?

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.

We say that the ground state for the particle in a box has nonzero energy. What goes wrong with Ψin equation 5.16 if n = 0 ?

In a study of heat transfer, we find that for a solid rod, there is a relationship between the second derivative of the temperature with respect to position along the rod and the first with respect to time. (A linear temperature change with position would imply as much heat flowing into a region as out. so the temperature there would not change with time).

∂2T(x,τ)∂x2=β∂T(x,τ)∂τδx

(a) Separate variables this assume a solution that is a product of a function of xand a function of tplug it in then divide by it, obtain two ordinary differential equations.

(b) consider a fairly simple, if somewhat unrealistic case suppose the temperature is 0 atx=0and, and x=1 positive in between, write down the simplest function of xthat (1) fits these conditions and (2) obey the differential equation involving x.Does your choice determine the value, including sign of some constant ?

(c) Obtain the fullT(x,t)for this case.
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