Chapter 5: Q6P (page 210)

Imagine two noninteracting particles, each of mass m, in the infinite square well. If one is in the state $ψ_{n}$ (Equation 2.28 ), and the other in state $ψ_{1} (l \neq n)$ , calculate localid="1658214464999" $<{(x_{1} - x_{2})}^{2}>$ , assuming (a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions.

Short Answer

Expert verified

a) The value of $< {(x_{1} - x_{2})}^{2} >$ assuming that they are distinguishable particles is $a^{2} [\frac{1}{6} - \frac{1}{2 π^{2}} (\frac{1}{n^{2}} + \frac{1}{m^{2}})]$ .

b) The value of $< {(x_{1} - x_{2})}^{2} >$ assuming that they are identical bosons is $a^{2} [\frac{1}{6} - \frac{1}{2 π^{2}} (\frac{1}{n^{2}} + \frac{1}{m^{2}})] - \frac{128 a^{2} m^{2} n^{2}}{π^{4} {(m^{2} - n^{2})}^{4}}$

c) The value of $< {(x_{1} - x_{2})}^{2} >$ assuming that they are identical fermions is $a^{2} [\frac{1}{6} - \frac{1}{2 π^{2}} (\frac{1}{n^{2}} + \frac{1}{m^{2}})] - \frac{128 a^{2} m^{2} n^{2}}{π^{4} {(m^{2} - n^{2})}^{4}}$ .

Step by step solution

Definition of identical bosons and identical fermions

According to Carroll, particles exist in two types: those that makeup matter, known as 'fermions,' and those that convey forces, known as 'bosons.

Bosons can be piled on top of one other, whereas fermions take up space.

(a) Determination of <(x1-x2)2> assuming that they are distinguishable particles

For distinguishable particles, use the following formula,

$<{(x_{1} - x_{2})}^{2}> = {<x^{2}>}_{a} + {<x^{2}>}_{b} - 2 {(x)}_{a} (x \tilde{n_{b}} = a^{2} [\frac{1}{3} - \frac{1}{2 {(ττn)}^{2}}] + a^{2} [\frac{1}{3} - \frac{1}{2 {(ττn)}^{2}}] - 2 \times \frac{a}{2} \times \frac{a}{2} = a^{2} [\frac{2}{3} - \frac{1}{2} - \frac{1}{2 {ττ}^{2}} (\frac{1}{n^{2}} + \frac{1}{m^{2}})] Evaluate the value of <{(x_{1} - x_{2})}^{2}> . <{(x_{1} - x_{2})}^{2}> = a^{2} [\frac{1}{6} - \frac{1}{2 {ττ}^{2}} (\frac{1}{n^{2}} + \frac{1}{m^{2}})] Thus, the value of <{(x_{1} - x_{2})}^{2}> assuming they are distinguishable particles is a^{2} [\frac{1}{6} - \frac{1}{2 {ττ}^{2}} (\frac{1}{n^{2}} + \frac{1}{m^{2}})] .$

(b) Determination of<x1-x22>assuming that they are the identical Bosons

For the same Bosons, the equation is as follows,

${<{(x_{1} - x_{2})}^{2}>}_{B} = {<x^{2}>}_{m} + {<x^{2}>}_{n} - 2 \overset{,}{a} x {\tilde{n}}_{n} \overset{,}{a} x {\tilde{n}}_{m} - 2 {|\overset{,}{a} x {\tilde{n}}_{n m}|}^{2} = {<{(x_{1} - x_{2})}^{2}>}_{d} - 2 {|\overset{,}{a} x {\tilde{n}}_{n m}|}^{2}$

But it is known that $\overset{,}{a} x {\tilde{n}}_{n m} = \frac{- 8 a m n}{τ τ^{2} {(m^{2} - n^{2})}^{2}}$ . So, that expression is as follows,

${|\overset{,}{a} x {\tilde{n}}_{n m}|}^{2} = {|\overset{,}{a} x {\tilde{n}}_{n m}|}^{2} = \overset{,}{a} x {\tilde{n}}_{n m} \overset{,}{a} x {\tilde{n}}_{n m} {<{(x_{1} - x_{2})}^{2}>}_{B} = a^{2} [\frac{1}{6} - \frac{1}{2 τ τ^{2}} (\frac{1}{n^{2}} + \frac{1}{m^{2}})] - \frac{128 a^{2} m^{2} n^{2}}{τ τ^{4} (m^{2} - n^{2})} Hence, the value of {<{(x_{1} - x_{2})}^{2}>}_{d} considering they are identical bosons is . a^{2} [\frac{1}{6} - \frac{1}{2 {ττ}^{2}} (\frac{1}{n^{2}} + \frac{1}{m^{2}})] - \frac{128 a^{2} m^{2} n^{2}}{{ττ}^{4} {(m^{2} - n^{2})}^{4}}$

(c) Determination of<x1-x22>assuming that they are the identical fermions

$For Fermions that are identical, the equation is as follows, {(x_{1} - x_{2})}_{f}^{2} = x_{a}^{2} + x_{b}^{2} - 2 \overset{,}{ax} {\tilde{n}}_{a} \overset{,}{ax} {\tilde{n}}_{b} + 2 {|\overset{,}{ax} {\tilde{n}}_{ab}|}^{2} = {(x_{1} - x_{2})}_{d}^{2} + 2 {|\overset{,}{ax} {\tilde{n}}_{ab}|}^{2} = a^{2} \frac{1}{6} - \frac{1}{2 {ττ}^{2}} (\frac{1}{n^{2}} + \frac{1}{m^{2}}) + \frac{128 a^{2} m^{2} n^{2}}{{ττ}^{4} {(m^{2} - n^{2})}^{4}} Thus, the value of {<{(x_{1} - x_{2})}^{2}>}_{d} when they are identical fermions is . = a^{2} \frac{1}{6} - \frac{1}{2 {ττ}^{2}} (\frac{1}{n^{2}} + \frac{1}{m^{2}}) + \frac{128 a^{2} m^{2} n^{2}}{{ττ}^{4} {(m^{2} - n^{2})}^{4}}$

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Short Answer

Step by step solution

Definition of identical bosons and identical fermions

(a) Determination of <(x1-x2)2> assuming that they are distinguishable particles

(b) Determination of<x1-x22>assuming that they are the identical Bosons

(c) Determination of<x1-x22>assuming that they are the identical fermions

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