The expectation value of the z coordinate, (z), in the ground state of the hydrogen atom (wave function: $${\psi _{100}}\left( r \right) = A{e^{ - \frac{r}{{{a_0}}}}},$$ where A is the normalization constant and $${a_0}$$ is the Bohr radius), is
The resonance widths $$\Gamma $$ of $$\rho ,\,\omega $$ and $$\phi $$ particle resonances satisfy the relation $${\Gamma _\rho } > {\Gamma _\omega } > {\Gamma _\phi }$$ . Their lifetimes r satisfy the relation
The spin' function of a free particle, in the basis in which sz is diagonal can be written as \[\left[ {\begin{array}{*{20}{c}}
1 \\
0
\end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}}
0 \\
1
\end{array}} \right]\] with eigen values \[ + \frac{\hbar }{2}\] and \[ - \frac{\hbar }{2}\] respectively. In the given basis, the normalized eigen function of sy with eigen value \[ - \frac{\hbar }{2}\] is
A one-dimensional harmonic oscillator is in the state $$\psi \left( x \right) = \frac{1}{{\sqrt {14} }}\left[ {3{\psi _0}\left( x \right) - 2{\psi _1}\left( x \right) + {\psi _2}\left( x \right)} \right],$$ where, $${\psi _0}\left( x \right),\,{\psi _1}\left( x \right)$$ and $${\psi _2}\left( x \right)$$ are the ground, first excited and second excited states, respectively. The probability of finding the oscillator in the ground state is
Let $$\overrightarrow {\bf{L}} $$ = (Lx, Ly, Lz) denotes the orbital angular momentum operators of a particle and let L+ = Lx + i Ly and L- = Lx - i Ly. The particle is in aneigen state of L2 and Lz eigen values $${\hbar ^2}\left( {l + 1} \right)$$ and $$\hbar l$$ respectively. The expectation value of L+L- in this state is
A one-dimensional harmonic oscillator carrying a charge -q is placed in a uniform electric field $$\overrightarrow {\bf{E}} $$ along the positive X-axis. The corresponding Hamiltonian operator is
An electron propagating along the X-axis passes through a slit of width ∆y = 1 nm. The uncertainty in the y-component of its velocity after passing through the slit is
