Chapter 4: Q29E (page 135)

In Example 4.2. neither ${|Ψ|}^{2}$ nor $|Ψ|$ 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, or $s^{-1}$ . 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 $| Ψ_{T} |$ at the center detector (interference maximum) in terms of the example’s given detection rate and b? (c) What would be $| Ψ_{1} |, {| Ψ_{1} |}^{2}$ , and the detection rate R at the center detector with one of the slits blocked?

Short Answer

Expert verified

a)Units of b is $s / m$

(b) The detection rate at the center detector and b is $10 \sqrt{b} s^{- 1 / 2}$

Step by step solution

Dimensional analysis.

(a)

The dimensional analysis comes to the rescue, as we are told that the units of measurement are $ψ^{2}$ is and $\frac{1}{m}$ that of R is $\frac{1}{s}$ . Since the relation between $ψ^{2}$ and R is $ψ^{2} = bR$

Therefore, the units of b are $s / m$

Detection rate.

(b)

In example 4.2, we were given that the detection rate (R) is 100, hence, we can directly find $ψ_{T}$ it from the relations given in part (a).

${| ψ_{1} |}^{2} = (100 b) s^{- 1}$

$| ψ_{1} | = (10 \sqrt{b}) s^{- 1 / 2}$

Step 3: Detection rate R.

(c)

The wave amplitude will be halved with just a slit open, thus we can find the new detection rate R using the relation derived in section b).

$\begin{matrix} | ψ_{1} | = \frac{1}{2} | ψ_{T} | \\ = \frac{1}{2} \times (10 \sqrt{b} s^{- 1 / 2}) \end{matrix}$

$\begin{array}{l} {| ψ_{1} |}^{2} = (25 b) s^{- 1} \\ {| ψ_{1} |}^{2} = R_{1} b \end{array}$

Comparing it we get,

$R_{1} = {25 s}^{-1}$

Result.

(a)Units of b is $s / m$

(b) The detection rate at the center detector and b is $10 \sqrt{b} s^{- 1 / 2}$

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Short Answer

Step by step solution

Dimensional analysis.

Detection rate.

Step 3: Detection rate R.

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