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. \(H^{-1/2}\)

The wavelength of a photon needed to remove a proton from a nucleus which is bound to the nucleus with \(1~\text{MeV}\) energy is nearly:
1. \(1.2~\text{nm}\)
2. \(1.2\times 10^{-3}~\text{nm}\)
3. \(1.2\times 10^{-6}~\text{nm}\)
4. \(1.2\times 10~\text{nm}\)

Consider a beam of electrons (each electron with energy \(E_0)\) incident on a metal surface kept in an evacuated chamber. Then:

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?

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

Photons absorbed in matter are converted to heat. A source emitting n photon/sec of frequency $\mathrm{\nu }$ is used to convert 1 kg of ice at $0°\mathrm{C}$ to water at $0°\mathrm{C}$. Then, the time T taken for the conversion:
 

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

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. \(\dfrac{P_e}{m_ec}=10^{-4}\)

Two particles \(A_1\) and \(A_2\) of masses \({m_1},m_2\)  \(({m_1>m_2})\) have the same de-Broglie wavelength. Then:

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?

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

2. (a), (d)

3. (c), (d)

4. (a), (b)

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

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 \(\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{\mathrm{i}}\)$({\mathrm{}}_{}$\(E_o\) = constant \(>0\)). Its de-Broglie wavelength at time \(t\) is given by: