A particle is dropped from a height $$H.$$ The de-Broglie wavelength of the particle as a function of height is proportional to:
1. $$H$$
2. $$H^{1/2}$$
3. $$H^{0}$$
4. $$H^{-1/2}$$

Subtopic:  De-broglie Wavelength |
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A particle moves in a closed orbit around the origin, due to a force which is directed towards the origin. The de-Broglie wavelength of the particle varies cyclically between two values $$\lambda_{1} , \lambda_{2}$$ with $$\lambda_{1} > \lambda_{2}$$. Which of the following statement/s is/are true?

 (a) The particle could be moving in a circular orbit with origin as the centre. (b) The particle could be moving in an elliptic orbit with origin as its focus. (c) When the de-Broglie wavelength is $$λ_1$$ , the particle is nearer the origin than when its value is $$λ_2$$. (d) When the de-Broglie wavelength is $$λ_2$$, the particle is nearer the origin than when its value is $$λ_1$$.

Choose the correct option:
1. (b), (d)
2. (a), (c)
3. (b), (c), (d)
4. (a), (c), (d)

Subtopic:  De-broglie Wavelength |
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The de-Broglie wavelength of a photon is twice the de-Broglie wavelength of an electron. The speed of the electron is $$v_e = \dfrac c {100}$$. Then,

1. $$\dfrac{E_e}{E_p}=10^{-4}$$
2. $$\dfrac{E_e}{E_p}=10^{-2}$$
3. $$\dfrac{P_e}{m_ec}=10^{-2}$$
4. $$\dfrac{P_e}{m_ec}=10^{-4}$$

Subtopic:  De-broglie Wavelength |
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Two particles $$A_1$$ and $$A_2$$ of masses $${m_1},m_2$$  $$({m_1>m_2})$$ have the same de-Broglie wavelength. Then:

 (a) their momenta (magnitude) are the same (b) their energies are the same (c) energy of $$A_1$$ is less than the energy of $$A_2$$ (d) energy of $$A_1$$ is more than the energy of $$A_2$$

Choose the correct option:
1. (b), (c)
2. (a), (c)
3. (c), (d)
4. (b), (d)
Subtopic:  De-broglie Wavelength |
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Relativistic corrections become necessary when the expression for the kinetic energy $$\dfrac{1}{2} ~\text{mv}^{2}$$, becomes comparable with $$\text{mc}^{2}$$, where m is the mass of the particle. At what de-Broglie wavelength, will relativistic corrections become important for an electron?

 (a) $$\lambda = 10 \text{ nm}$$ (b) $$\lambda = 10^{- 1} \text{ nm}$$ (c) $$\lambda = 10^{- 4} \text{ nm}$$ (d) $$\lambda = 10^{- 6} \text{ nm}$$

Choose the correct option:
1. (a), (c)

2. (a), (d)

3. (c), (d)

4. (a), (b)

Subtopic:  De-broglie Wavelength |
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An electron (mass m) with an initial velocity $$\overset{\rightarrow}{v} = v_{0} \hat{i}$$ is in an electric field $$\overset{\rightarrow}{E} = E_{0} \hat{j}$$. If $$\lambda_{0} = \dfrac{h}{ {mv}_0}$$, its de-Broglie wavelength at time t is given by:

1. $$\lambda_0$$

2. $$\lambda_{0} \sqrt{1 + \dfrac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}$$

3. $$\dfrac{\lambda_{0}}{\sqrt{1 + \dfrac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}$$

4. $$\dfrac{\lambda_{0}}{\left(1 + \dfrac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}\right)}$$

Subtopic:  De-broglie Wavelength |
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An electron (mass $$m$$) with an initial velocity $$\overrightarrow{\mathrm{v}}=\mathrm{v}_0 \hat{\mathrm{i}}$$ $\stackrel{}{\mathrm{}}$$$(\mathrm{v}_0>0)$$ is in an electric field $$\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{\mathrm{i}}$$$\left({\mathrm{}}_{}$$$E_o$$ = constant $$>0$$). Its de-Broglie wavelength at time $$t$$ is given by:

 1 $$\dfrac{\lambda_0}{\left(1+\dfrac{e E_0}{m} \dfrac{t}{\mathrm{v}_0}\right)}$$ 2 $$\lambda_0\left(1+\dfrac{e E_0 t}{m \mathrm{v}_0}\right)$$ 3 $$\lambda_0$$ 4 $$\lambda_0t$$
Subtopic:  De-broglie Wavelength |
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An electron is moving with an initial velocity $\stackrel{\to }{\mathrm{v}}={\mathrm{v}}_{0}\stackrel{^}{\mathrm{i}}$ and is in a magnetic field $\stackrel{\to }{\mathrm{B}}={\mathrm{B}}_{0}\stackrel{^}{\mathrm{j}}$. Then, its de-Broglie wavelength:

1. remains constant

2. increases with time

3. decreases with time

4. increases and decreases periodically

Subtopic:  De-broglie Wavelength |
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A proton, a neutron, an electron and an $\mathrm{\alpha }$-particle have the same energy. Then, their de-Broglie wavelengths compare as:

1. ${\mathrm{\lambda }}_{\mathrm{p}}={\mathrm{\lambda }}_{\mathrm{n}}>{\mathrm{\lambda }}_{\mathrm{e}}>{\mathrm{\lambda }}_{\mathrm{\alpha }}$

2. ${\mathrm{\lambda }}_{\mathrm{\alpha }}<{\mathrm{\lambda }}_{\mathrm{p}}={\mathrm{\lambda }}_{\mathrm{n}}<{\mathrm{\lambda }}_{\mathrm{e}}$

3. ${\mathrm{\lambda }}_{\mathrm{e}}<{\mathrm{\lambda }}_{\mathrm{p}}={\mathrm{\lambda }}_{\mathrm{n}}>{\mathrm{\lambda }}_{\mathrm{\alpha }}$

4. ${\mathrm{\lambda }}_{\mathrm{e}}={\mathrm{\lambda }}_{\mathrm{p}}={\mathrm{\lambda }}_{\mathrm{n}}={\mathrm{\lambda }}_{\mathrm{\alpha }}$

Subtopic:  De-broglie Wavelength |
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