Dual Nature of Radiation and Matter Test

Directions For Questions

When a particle is restricted to move along 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 $$E = p^{2}/2m$$. Thus, the energy of the particle can be denoted by a quantum number n taking values $$1, 2, 3, . $$($$n = 1$$, 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 $$h = 6.6 \times 10^{-34} Js$$ and $$e = 1.6 \times 10^{-19} C.$$

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Dual Nature of ...

Question 1

1 photon with energy $$10.2 eV$$ and an electron with energy $$1.4 eV$$

2 photon with energy $$10.2 eV$$

2 photon with energy $$1.4 eV$$

One photon with energy $$3.4 eV$$ and 1 electron with energy $$1.4 eV$$

Question 2

$$\mathrm{a}^{-2}$$

$$\mathrm{a}^{-3/2}$$

$$\mathrm{a}^{-1}$$

$$\mathrm{a}^{2}$$

Question 3

$$\displaystyle { \lambda }_{ e }$$ increases at the same rate as $${ \lambda }_{ ph }$$ for $$\lambda_{ph} < hc / \phi$$

$$\displaystyle { \lambda }_{ e }$$ is approximately halved, if d is doubled

$$\displaystyle { \lambda }_{ e }$$ decreases with increase in $$\displaystyle \phi $$ and $$\displaystyle { \lambda }_{ ph }$$

For large potential difference $$\displaystyle \left( V>>\phi /e \right) { \lambda }_{ e }$$ is approximately halved if V is made four time

Question 4

$$\leq 2.8 \times 10^{-12}m$$

$$< 2.8 \times 10^{-10}m$$

$$< 2.8 \times 10^{-9}m$$

$$\geq 2.8 \times 10^{-9}m$$

Question 5

$$400P$$

$$\cfrac{P}{200}$$

$$100P$$

$$200P$$

Question 6

$$\lambda _{p}\propto \lambda _{e}^{2}$$

$$\lambda _{p}\propto \lambda _{e}$$

$$\lambda _{p}\propto \sqrt{\lambda _{e}}$$

$$\displaystyle \lambda _{p}\propto \dfrac{1}{\sqrt{\lambda _{e}}}$$

Question 7

$$\dfrac { 1 }{ c } { \left( \dfrac { E }{ 2m } \right) }^{ \dfrac { 1 }{ 2 } }$$

$${ \left( \dfrac { E }{ 2m } \right) }^{ \dfrac { 1 }{ 2 } }$$

$$c{ \left( 2mE \right) }^{ \dfrac { 1 }{ 2 } }$$

$$\dfrac { 1 }{ c } { \left( \dfrac { 2m }{ E } \right) }^{ \dfrac { 1 }{ 2 } }$$($$c$$ being velocity of light)

Question 8

$$10^2V$$

$$10^3V$$

$$10^4V$$

$$10V$$

Question 9

25

75

60

50

Question 10

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$$\dfrac { 1 }{ c } { \left( \dfrac { 2m }{ E } \right) }^{ \dfrac { 1 }{ 2 } }$$

($$c$$ being velocity of light)