The speed of the particle, that can take discrete values, is proportional to [2009]

Question

When a particle is restricted to move along the $x$ -axis between $x = 0$ and $x = a$ , where $a$ is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends $x = 0$ and $x = a$ . The wavelength of this standing wave is related to the linear momentum $p$ of the particle according to the de Broglie relation. The energy of the particle of mass $m$ is related 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 values $1, 2, 3, \dots$ ( $n = 1$ , called the ground state) corresponding to the number of loops in the standing wave.

Use the model described above to answer the following three questions for a particle moving in the line $x = 0$ to $x = a$ . Take $h = 6.6 \times 10^{- 34} J s$ and $e = 1.6 \times 10^{- 19} C.$

Q. The speed of the particle, that can take discrete values, is proportional to [2009]

Use the model described above to answer the following three questions for a particle moving in the line $x = 0$ to $x = a$ . Take $h = 6.6 \times 10^{- 34} J s$ and $e = 1.6 \times 10^{- 19} C.$

Q. The speed of the particle, that can take discrete values, is proportional to [2009]

Smart Achievers · Accepted Answer

(4)

$λ = \frac{h}{p}$ $\Rightarrow λ = \frac{h}{m v}$

But, $\frac{n λ}{2} = a$ $m$

$∴$ $\frac{h}{m v} = \frac{2 a}{n}$

$∴$ $m v = \frac{n h}{2 a}$ $\Rightarrow v = \frac{n h}{2 a m}$

$∴$ $v \propto n$

Solution

1 $n^{- 3 / 2}$

2 $n^{- 1}$

3 $n^{1 / 2}$

4 $n$

Ans.
(4)

$λ = \frac{h}{p}$ $\Rightarrow λ = \frac{h}{m v}$

But,   $\frac{n λ}{2} = a$ $\Rightarrow λ = \frac{2 a}{n}$

$∴$    $\frac{h}{m v} = \frac{2 a}{n}$

$∴$    $m v = \frac{n h}{2 a}$ $\Rightarrow v = \frac{n h}{2 a m}$

$∴$    $v \propto n$

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Solution

1 n-3/2

2 n-1

3 n1/2

4 n

Ans. (4) λ=hp⇒λ=hmv But, nλ2=a⇒λ=2an ∴ hmv=2an ∴ mv=nh2a⇒v=nh2am ∴ v∝n

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1 $n^{- 3 / 2}$

2 $n^{- 1}$

3 $n^{1 / 2}$

4 $n$

Ans.
(4)

$λ = \frac{h}{p}$ $\Rightarrow λ = \frac{h}{m v}$

But, $\frac{n λ}{2} = a$ $\Rightarrow λ = \frac{2 a}{n}$

$∴$ $\frac{h}{m v} = \frac{2 a}{n}$

$∴$ $m v = \frac{n h}{2 a}$ $\Rightarrow v = \frac{n h}{2 a m}$

$∴$ $v \propto n$