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Chapter 4: Q46E (page 137)

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.

Short Answer

Expert verified

The minimum kinetic energy formula isK·E.≥h28mL2

Step by step solution

01

Uncertainty principle.

That the position and the velocity of an object cannot both be measured exactly, at the same time, even in theory.

Δx∆p≥h4π

Δxuncertainty in position.

Δpuncertainty of momentum.

h Planck's constant.

02

Kinetic energy.

ΔxΔp=ΔxΔ(mv)⩾h2

localid="1659187305509" Δv=h2mΔx

Here Δx=L

Δv=h2mL

The kinetic energyK.E.=12m(Δv)2

Substitute the value,

localid="1659187247106" K.E.≥h28mL2

Therefore, the minimum Kinetic Energy formula of the width is localid="1659187237799" K.E≥.h28mL2

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