Quantum Mechanics-I

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Chapter#6: Three-Dimensional Momentum

Team Quanta presents all possible short questions of BS Physics book Quantum Mechanics-I’s Chapter#6: Three-Dimensional Momentum.

Q.8.1 Write Eigen value equation for a composite particle in a central potential

    \[\mathbf{V\left(r\right)=-\frac{e^2}{r}}\]

Answer: Hamiltonian of system is,

    \[H=- \frac{\hbar^2}{2\mu}\ \frac{1}{r}\ \frac{\partial^2}{{\partial r}^2}\left(r\right)+\frac{L^2}{{2\mu r}^2}+V\left(r\right)\]

Eigen value equation (Schrodinger equation) is,

    \[\left { - \frac{\hbar}{2\mu}\frac{1}{r}\ \frac{\partial^2}{{\partial r}^2}\left(r\right)+\frac{L^2}{{2\mu r}^2}-\frac{e^2}{r} \right } \mathrm{\Psi}\left(r,\ \theta,\ \varphi\right)=E\mathrm{\Psi}\left(r,\ \theta,\ \varphi\right)\]

Hence \mu is reduced mass.

Q.8.2 A hydrogen atom can be viewed as two particle system – a proton and an electron with Coulomb interacting potential between them. Write the Schrodinger wave equation for such a system by explaining all the terms used.

Answer: Schrodinger wave equation for proton and electron is,

    \[\left{-\frac{\hbar}{2}\left(\frac{\nabla_1^2}{m_p}+\frac{\nabla_2^2}{m_e}\right)+V\left(r\right)\right}\mathrm{\Psi}\ =E\mathrm{\Psi}\]

mp is mass of proton and me is mass of electron. The indices 1 & 2 indicate proton and electron respectively.

Q.8.3 what is rigid rotator? Give at least its one application.

Answer: A system of two spherical masses rigidly located at a finite fixed distance from one another and capable free to rotate about its center of mass is called a rigid rotator with free axis. An example of it is a diatomic molecule with two atoms at some distance apart because the molecule can freely rotate about its center of gravity. The theory of rigid rotator will, therefore, lead to interpretation of spectra of diatomic molecules.

Q.8.4 Radial distribution function is given by,

    \[P\left(r\right)=\frac{4}{a_0^3}r^2e^{-\frac{2r}{a_0}}\]

Find the most probable distance.

Answer: For the most probable distance r, P(r) should be maximum, so

    \[\frac{dP}{dr}=0\ \ \ \ \ \ \Rightarrow\frac{d}{dr}\left(\frac{4}{a_0^3}r^2e^\frac{-2r}{a_0}\right)=0\ \ \ \ \ \ \ \Rightarrow r=a_0\]

Thus the most probable distance of the electron from the nucleus in the normal state of hydrogen atom is equal to Bohr’s radius.

Q.8.5 The energy of harmonic oscillator in three dimensions is given by,

    \[E=\left(n_x+\frac{1}{2}\right)\hbar \omega_x+\left(n_y+\frac{1}{2}\right) \hbar\omega_y+\left(n_z+\frac{1}{2}\right)\hbar \omega_z\]

Find energy of isotropic oscillator. Also find degeneracy of ground state.

Answer: For isotropic oscillator,

    \[\omega_x=\omega_y=\omega_z=\omega\]

So, energy Eigen values of isotropic oscillator are,

    \[E_{n_xn_yn_z}=\left(n_x+n_y+n_z+\frac{3}{2}\right) \hbar \omega\]

Ground state energy is,

    \[E_{000}=\frac{3}{2} \hbar \omega\]

This shows that ground state is non-degenerate.

Q.8.6 The wave function of an electron in a hydrogen like atom is,

    \[\Psi=De^\frac{-r}{a}\]

Find normalization constant D.

Answer: Using normalization condition,

    \[\left|D\right|^2\int_{0}^{\infty}{r^2e^\frac{-2r}{a}dr\times\int_{0}^{\pi}{sin\theta d\theta\times\int_{0}^{2\pi}{d\varphi=1}}}\]

    \[\Rightarrow\left|D\right|^2\frac{a^3}{4}\times2\times2\pi=1\ \ \ \ \ \ \ \Rightarrow D=\frac{1}{\sqrt{\pi a^3}}\]

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