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Chapter 4: Q37E (page 136)

A free particle is represented by the plane wave function蠄(x,t)=Aexp[i(1.58脳1012x-7.91脳1016t)]where SIunits are understood. What are the particle鈥檚 momentum, Kinetic energy, and mass? (Note: In nonrelativistic quantum mechanics, we ignore mass/internal energy, so the frequency is related to kinetic energy alone.)

Short Answer

Expert verified

Particle鈥檚 Momentum of Plane Wave Function 1.66x10-22 kg.m/s

Particle鈥檚 Momentum of Kinetic Energy 8.30x10-18 J

Particle鈥檚 Momentum of Mass 1.66x10-27s

Step by step solution

01

Given information

The given wavefunction is蠄(x,t)=Aexp[i(1.58脳1012x-7.91脳1016t)]

02

Concept Introduction

The equation of an electromagnetic wave can be expressed as,

蠄(x,t)=Aei(kx-蝇t)鈥︹赌︹赌︹赌�(1)

03

Calculations for momentum

Compare the given wavefunction with equation (1), and we get

k=1.58脳1012m-1蝇=7.91脳1016s-1

We use the following formula to find the momentum, p :

p=hk=1.05脳10-34J路s脳(1.58脳1012m-1)=1.66脳10-22kg路m/s

04

Calculate the Kinetic energy.

To calculate the kinetic energy, , we use:

KE=h蝇=1.05脳10-34J路s脳7.91脳1016s-1=8.30脳10-18J

05

Calculate the mass.

For calculation of the Mass , we use:

m=p22脳KE

Substitute the obtained values in the above expression and we get,

m=1.66脳10-22kg路m/s22脳8.3脳10-18J=1.66脳10-27s

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Most popular questions from this chapter

One of the cornerstones of quantum mechanics is that bound particles cannot be stationary-even at zero absolute temperature! A "bound" particle is one that is confined in some finite region of space. as is an atom in a solid. There is a nonzero lower limit on the kinetic energy of such a particle. Suppose minimum kinetic energy of width L. Obtain an approximate formula for its minimum kinetic energy.

In the hydrogen atom, the electron鈥檚 orbit, not necessarily circular, extends to a distance of a about an angstrom (1脜=0.1鈥塶尘)from the proton. If it is to move about as a compact classical particle in the region where it is confined, the electron鈥檚 wavelength had better always be much smaller than an angstrom. Here we investigate how large might be the electron鈥檚 wavelength. If orbiting as a particle, its speed at 1脜could be no faster than that for circular orbit at that radius. Why? Find the corresponding wavelength and compare it to1脜 . Can the atom be treated classically?

Determine the momenta that can never be measured when a particle has a wave function.

蠄(x)={C;|x|鈮�+12w0;|x|>12w}

Classically and nonrelativistically, we say that the energyEof a massive free particle is just its kinetic energy. (a) With this assumption, show that the classical particle velocityvparticleis2E/p. (b) Show that this velocity and that of the matter wave differ by a factor of 2. (c) In reality, a massive object also has internal energy, no matter how slowly it moves, and its total energyEis纬mc2, where纬=1/1-(vparticle/c)2.Show thatvparticleispc2/Eand thatvwaveisc2/vparticleIs there anything wrong with itvwave? (The issue is discussed further in Chapter 6.)

Experiments effectively equivalent to the electron double slit have been conducted in different, novel ways, producing obvious maxima and minima. Often the point is stressed that the intensity is extremely low. Why is this fact emphasized so much? How low is low enough to make the point?
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