61. A classical particle is moving in an external potential field V(x, y, z) which is invariant under the following infinitesimal transformations
\[\begin{array}{*{20}{c}}
{x \to x'}& = &{x + \delta x} \\
{y \to y'}& = &{y + \delta y} \\
{\left[ {\begin{array}{*{20}{c}}
x \\
y
\end{array}} \right] \to \left[ {\begin{array}{*{20}{c}}
{x'} \\
{y'}
\end{array}} \right]}& = &{{R_z}\left[ {\begin{array}{*{20}{c}}
x \\
y
\end{array}} \right]}
\end{array}\]
where, Rz is the matrix corresponding to rotation about the Z-axis. The conserved quantities are (the symbols have their usual meaning)
\[\begin{array}{*{20}{c}} {x \to x'}& = &{x + \delta x} \\ {y \to y'}& = &{y + \delta y} \\ {\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] \to \left[ {\begin{array}{*{20}{c}} {x'} \\ {y'} \end{array}} \right]}& = &{{R_z}\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right]} \end{array}\]
where, Rz is the matrix corresponding to rotation about the Z-axis. The conserved quantities are (the symbols have their usual meaning)
62. An atom emits a photon of wavelength λ = 600 nm by transition from an excited state of life time 8 × 10-9 s. If $$\Delta \nu $$ represents the minimum uncertainty in the frequency of the photon, the fractional width $$\frac{{\Delta \nu }}{\nu }$$ of the spectral line is of the order of
63. A particle of mass m is confined in the potential
\[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
{\frac{1}{2}m{\omega ^2}{x^2},}&{{\text{for }}x < 0} \\
{\infty ,}&{{\text{for }}x \leqslant 0}
\end{array}} \right.\]
Let the wave function of the particle be given by $$\psi \left( x \right) = - \frac{1}{{\sqrt 5 }}{\psi _0} + \frac{2}{{\sqrt 5 }}{\psi _1}$$
where $${\psi _0}$$ and $${\psi _1}$$ are the eigen functions of the ground state and the first excited slate respectively. The expectation value of the energy is
\[V\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{2}m{\omega ^2}{x^2},}&{{\text{for }}x < 0} \\ {\infty ,}&{{\text{for }}x \leqslant 0} \end{array}} \right.\]
Let the wave function of the particle be given by $$\psi \left( x \right) = - \frac{1}{{\sqrt 5 }}{\psi _0} + \frac{2}{{\sqrt 5 }}{\psi _1}$$
where $${\psi _0}$$ and $${\psi _1}$$ are the eigen functions of the ground state and the first excited slate respectively. The expectation value of the energy is
64. The normalized ground state, wave function of a hydrogen atom is given $$\psi \left( r \right) = \frac{1}{{\sqrt {4\pi } }}\frac{2}{{{a^{\frac{3}{2}}}}} - {e^{ - \frac{r}{a}}},$$ where a is the Bohr radius and r is the distance of the electron from the nucleus located at the origin. The expectation value $$\left\langle {\frac{1}{{{r^2}}}} \right\rangle $$ is
65. A free particle is moving in +X direction with a linear momentum p. The wave function of the particle normalised in a length L is
66. The wave function of a one-dimensional harmonic oscillator is $${\psi _0} = A\exp \left( {\frac{{ - {\alpha ^2}{x^2}}}{2}} \right)$$ for the ground state $${E_0} = \frac{{\hbar \omega }}{2},$$ where $${\alpha ^2} = \frac{{m\omega }}{\hbar }$$ in the presence of a perturbing potential of $${E_0}{\left( {\frac{{\alpha x}}{{10}}} \right)^4},$$ the first order Change in the ground state energy is
$$\left[ {{\text{Given, }}\Gamma \left( {x + 1} \right) = \int_0^\infty {{t^x}\exp \left( { - t} \right)dt} } \right]$$
$$\left[ {{\text{Given, }}\Gamma \left( {x + 1} \right) = \int_0^\infty {{t^x}\exp \left( { - t} \right)dt} } \right]$$
67. For a spin $$\frac{1}{2}$$ particle, the expectation value of sxsysz, where sx, sy and sz are spin operators, is
68. A free particle with energy E whose wave function is a plane wave with wavelength λ enters a region of constant potential V > 0, where the wavelength of the particle is 2λ. The ratio (V/E) is
69. A one-dimensional random walker takes steps to left or right with equal probability. The probability that the random walker starting from origin is back to origin after N even number of steps is
70. An electron with energy E is incident from left on a potential barrier, given by \[\begin{gathered}
V\left( x \right) = 0{\text{ for }}x < 0 \hfill \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {V_0}{\text{ for }}x > 0 \hfill \\
\end{gathered} \] as shown in the figure.
For E < V0, the space part of the wave function for x > 0 is of the form
For E < V0, the space part of the wave function for x > 0 is of the form