When a particle is restricted to move along x- axis between $x = 0 and x = 4$ whwre a is opf nanometer demension , its energy can take only certain spscfic values . The allowed energies of the particles only in such a restiricted regain , correspond to the formation of standing wave with nodes at its end $x = 0 and x = a$ .The wavelength of this standing wave is related to the linear momentum p of the paarticle according to the de Broglie relation .The energy of the particle of mass $m$ is reated to its linear momentum as
$E = \frac{p ^{2}}{2 m}$ . thus , the energy of the particle can be denoted by a quantum number $n$ taking value $1, 2, 3, \dots . (n = 1, c a l \leq d t h e g r o u n d s t a t e)$ corresponding to the number of loops in the standing wave use the model described above to answer the following there question for a particle moving in the line $x = 0 \to x = a T a k e h = 6.6 \times 10^{- 34} J s and e = 1.6 \times 10^{- 19} C$
The alloewd energy for the particle for a particular value of $n$ is proportional to

Question

When a particle is restricted to move along x- axis between

x = 0 and x = 4

whwre a is opf nanometer demension , its energy can take only certain spscfic values . The allowed energies of the particles only in such a restiricted regain , correspond to the formation of standing wave with nodes at its end

x = 0 and x = a

.The wavelength of this standing wave is related to the linear momentum p of the paarticle according to the de Broglie relation .The energy of the particle of mass

m

is reated to its linear momentum as

E = \frac{p ^{2}}{2 m}

. thus , the energy of the particle can be denoted by a quantum number

n

taking value

1, 2, 3, \dots . (n = 1, c a l \leq d t h e g r o u n d s t a t e)

corresponding to the number of loops in the standing wave use the model described above to answer the following there question for a particle moving in the line

x = 0 \to x = a T a k e h = 6.6 \times 10^{- 34} J s and e = 1.6 \times 10^{- 19} C

The alloewd energy for the particle for a particular value of

n

is proportional to

Filo · Accepted Answer

Answer: ASolution: λ=ph​andE=2mp2​   rArr E = (h^(2))/(2m lambda^(2))The≤n>h∈whichthepartic≤isrestrictedismoveisaThis≤n>hisaμ<ip≤of(lambda)/(2)Now,n(lambda)/(2) = u rArr lambda = (2a)/(n)rArr E = ((h^(2) n^(2))/(2 m xx 4a^(2)) = (n^(2) h^(2))/(8 ma^(2))⇒E∝a−2 for a particular value of n

Question Text	When a particle is restricted to move along x- axis between $x = 0 and x = 4$ whwre a is opf nanometer demension , its energy can take only certain spscfic values . The allowed energies of the particles only in such a restiricted regain , correspond to the formation of standing wave with nodes at its end $x = 0 and x = a$ .The wavelength of this standing wave is related to the linear momentum p of the paarticle according to the de Broglie relation .The energy of the particle of mass $m$ is reated to its linear momentum as $E = \frac{p ^{2}}{2 m}$ . thus , the energy of the particle can be denoted by a quantum number $n$ taking value $1, 2, 3, \dots . (n = 1, c a l \leq d t h e g r o u n d s t a t e)$ corresponding to the number of loops in the standing wave use the model described above to answer the following there question for a particle moving in the line $x = 0 \to x = a T a k e h = 6.6 \times 10^{- 34} J s and e = 1.6 \times 10^{- 19} C$ The alloewd energy for the particle for a particular value of $n$ is proportional to
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