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Chapter 4: Q9CQ (page 133)

Question: A classmate studies Figures 12 and 17, then claims that when a spot appears, its location simultaneously establishes the particle’s -component of momentum, according to the angle from center, and its position (i.e., at the spot). How do you answer this claim?

Short Answer

Expert verified

Answer:

We are unable to concurrently measure location and momentum because the detecting procedure changes the momentum of the particle (uncertainty principle).

Step by step solution

01

Uncertainty relation

Mathematically, the uncertainty relation can be stated asΔxΔp⩾h2 .

02

Explanation 

The Heisenberg uncertainty principle is a basic tenet of nature and is not only an issue of experimental scope. You have just altered the particle's momentum as a result of the detecting procedure as the particle position on the detector is determined.

Therefore, it is impossible to estimate the location without affecting the particle's momentum. We always have a limited uncertainty associated with the position measurement since it doesn't provide us with the precise location of the particle.

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Most popular questions from this chapter

Verify the claim made in a section 4.4 that if all results of a repeated experiment are equal, the standard deviation, equation (4.13) Will be0.

In the Bohr model of the hydrogen atom, the electron can have only certain velocities. Obtain a formula for the allowed velocities, then obtain a numerical value for the highest speed possible.

In Exercise 45, the case is made that the position uncertainty for a typical macroscopic object is generally so much smaller than its actual physical dimensions that applying the uncertainty principle would be absurd. Here we gain same idea of how small an object would have♦ to be before quantum mechanics might rear its head. The density of aluminum is 2.7×103 kg/m3, is typical of solids and liquids around us. Suppose we could narrow down the velocity of an aluminum sphere to within an uncertainty of1μmper decade. How small would it have to be for its position uncertainty to be at least as large as110%of its radius?

In Example 4.2. neither|Ψ|2nor|Ψ|are given units—only proportionalities are used. Here we verify that the results are unaffected. The actual values given in the example are particle detection rates, in particles/second, ors-1. For this quantity, let us use the symbol R. It is true that the particle detection rate and the probability density will be proportional, so we may write|Ψ|2= bR, where b is the proportionality constant. (b) What must be the units of b? (b) What is|ΨT|at the center detector (interference maximum) in terms of the example’s given detection rate and b? (c) What would be|Ψ1|,|Ψ1|2, and the detection rate R at the center detector with one of the slits blocked?

What is the density of a solid'? Although the mass densities of solids vary greatly, the number densities (in mol/m3) vary surprisingly little. The value. of course, hinges on the separation between the atoms-but where does a theoretical prediction start? The electron in hydrogen must not as a particle orbiting at a strict radius, as in the Bohr atom. but as a diffuse orbiting wave. Given a diffuse probability, identically repeated experiments dedicated to "finding" the electron would obtain a range of values - with a mean and standard deviation (uncertainty). Nevertheless, the allowed radii predicted by the Bohr model are very close to the true mean values.

(a) Assuming that atoms are packed into a solid typically jBohr radii apart, what would be the number of moles per cubic meter?

(b) Compare this with the typical mole density in a solid of105mol/m3. What would be the value of j?
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