CSIR NET Physical Sciences Quantum Mechanics Syllabus

Quantum Mechanics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Quantum Mechanics in CSIR NET Physical Sciences, not a summary of it.

1. 1

   Wave-particle duality

   1 topic

   • Introduction to wave-particle duality
2. 2

   Schrödinger equation

   2 topics

   • Time-dependent Schrödinger equation
   • Time-independent Schrödinger equation
3. 3

   Eigenvalue problems

   2 topics

   • Particle in a box
   • Harmonic oscillator
4. 4

   Tunneling through a barrier

   1 topic

   • Introduction to tunneling
5. 5

   Wave-function representations

   2 topics

   • Coordinate representation
   • Momentum representation
6. 6

   Commutators and Heisenberg uncertainty principle

   2 topics

   • Introduction to commutators
   • Heisenberg uncertainty principle
7. 7

   Dirac notation for state vectors

   1 topic

   • Introduction to Dirac notation
8. 8

   Motion in a central potential

   4 topics

   • Orbital angular momentum
   • Angular momentum algebra
   • Spin
   • Addition of angular momenta
9. 9

   Hydrogen atom

   1 topic

   • Introduction to Hydrogen atom
10. 10

    Stern-Gerlach experiment

    1 topic

    • Introduction to Stern-Gerlach experiment
11. 11

    Time-independent perturbation theory

    1 topic

    • Introduction to perturbation theory
12. 12

    Variational method

    1 topic

    • Introduction to variational method
13. 13

    Time dependent perturbation theory

    1 topic

    • Introduction to time-dependent perturbation theory
14. 14

    Fermi's golden rule

    1 topic

    • Introduction to Fermi's golden rule
15. 15

    Selection rules

    1 topic

    • Introduction to selection rules
16. 16

    Identical particles

    1 topic

    • Introduction to identical particles
17. 17

    Pauli exclusion principle

    1 topic

    • Introduction to Pauli exclusion principle
18. 18

    Spin-statistics connection

    1 topic

    • Introduction to spin-statistics connection

Quantum Mechanics flashcards for CSIR NET Physical Sciences

22 of 50 cards from the Quantum Mechanics deck — real questions with worked answers.

1. In partial wave analysis, how is the scattering amplitude $f(\theta)$ expressed as a sum over phase shifts $\delta_l$?

   $$f(\theta)=\frac{1}{k}\sum_{l=0}^{\infty}(2l+1)e^{i\delta_l}\sin\delta_l\,P_l(\cos\theta)$$ where $k$ is the wave number and $P_l$ are Legendre polynomials.

2. What is the partial-wave expression for the total scattering cross-section in terms of phase shifts?

   $$\sigma=\frac{4\pi}{k^{2}}\sum_{l=0}^{\infty}(2l+1)\sin^{2}\delta_l$$

3. State the optical theorem relating the total cross-section to the forward scattering amplitude.

   $$\sigma_{\text{tot}}=\frac{4\pi}{k}\,\text{Im}\,f(0)$$ The total cross-section is proportional to the imaginary part of the forward ($\theta=0$) scattering amplitude.

4. Physically, what is a phase shift $\delta_l$ in scattering theory?

   It is the phase by which the $l$-th partial radial wave is shifted (relative to the free-particle solution) due to the scattering potential. The asymptotic radial wave behaves as $\sin\!\left(kr-\frac{l\pi}{2}+\delta_l\right)$.

5. How does the sign of the phase shift $\delta_l$ distinguish attractive from repulsive potentials?

   A positive phase shift ($\delta_l>0$) corresponds to an attractive potential (wave pulled inward), while a negative phase shift ($\delta_l<0$) corresponds to a repulsive potential (wave pushed outward).

6. What is the partial-wave contribution $\sigma_l$ to the cross-section, and what is its maximum value (unitarity limit)?

   $$\sigma_l=\frac{4\pi}{k^{2}}(2l+1)\sin^{2}\delta_l$$ Its maximum (when $\delta_l=\pi/2$) is $\sigma_l^{\max}=\frac{4\pi}{k^{2}}(2l+1)$, the unitarity limit.

7. What is the partial-wave $S$-matrix element $S_l$ in terms of the phase shift?

   $$S_l=e^{2i\delta_l}$$ For purely elastic scattering $|S_l|=1$ (unitarity).

8. At low energies (long wavelength), which partial wave dominates the scattering, and why?

   The $s$-wave ($l=0$) dominates. Higher partial waves require the particle to have angular momentum $l\hbar\approx pb$, but at low $k$ the impact parameter $b\sim l/k$ exceeds the range of the potential, so $\delta_l\to0$ for $l\geq1$.

9. Define the scattering length $a$ in terms of the low-energy $s$-wave phase shift.

   $$a=-\lim_{k\to0}\frac{\delta_0}{k}$$ The low-energy $s$-wave cross-section is $\sigma_0=4\pi a^{2}$.

10. What is the low-energy ($k\to0$) limit of the total cross-section for $s$-wave scattering with scattering length $a$?

    $$\sigma=4\pi a^{2}$$ It becomes isotropic and energy-independent.

11. How does the phase shift $\delta_l$ behave with energy near a resonance (Breit-Wigner form)?

    $\delta_l$ rises rapidly through $\pi/2$ as the energy passes the resonance energy $E_R$. Near resonance $\tan\delta_l=\frac{\Gamma/2}{E_R-E}$, giving a peak in $\sin^{2}\delta_l$.

12. State Levinson's theorem relating the zero-energy phase shift to bound states.

    $$\delta_l(0)-\delta_l(\infty)=n_l\,\pi$$ where $n_l$ is the number of bound states of angular momentum $l$ supported by the potential.

13. How is the differential cross-section related to the scattering amplitude?

    $$\frac{d\sigma}{d\Omega}=|f(\theta)|^{2}$$

14. Write the asymptotic form of the scattering wave function used to define the scattering amplitude.

    $$\psi(\vec r)\xrightarrow{r\to\infty}e^{ikz}+f(\theta)\frac{e^{ikr}}{r}$$ an incident plane wave plus an outgoing spherical wave.

15. What is the first Born approximation for the scattering amplitude in terms of the potential $V(\vec r)$?

    $$f^{(B)}(\theta)=-\frac{m}{2\pi\hbar^{2}}\int e^{i\vec q\cdot\vec r}\,V(\vec r)\,d^{3}r$$ where $\vec q=\vec k_i-\vec k_f$ is the momentum transfer; $f^{(B)}$ is essentially the Fourier transform of the potential.

16. In the Born approximation, what is the magnitude of the momentum transfer $q$ in terms of $k$ and the scattering angle $\theta$?

    $$q=2k\sin\!\left(\frac{\theta}{2}\right)$$

17. For a spherically symmetric potential, what is the simplified Born-approximation formula for $f(\theta)$?

    $$f^{(B)}(\theta)=-\frac{2m}{\hbar^{2}q}\int_{0}^{\infty} r\,V(r)\,\sin(qr)\,dr$$

18. Under what physical condition is the (first) Born approximation valid?

    When the scattering potential is weak and/or the energy is high, so the incident wave is only slightly perturbed. Quantitatively the scattered wave inside the potential region must be small compared with the incident wave.

19. What is the Born-approximation differential cross-section for the screened (Yukawa) potential $V(r)=V_0\,\frac{e^{-\mu r}}{r}$?

    $$\frac{d\sigma}{d\Omega}=\left(\frac{2mV_0}{\hbar^{2}}\right)^{2}\frac{1}{(\mu^{2}+q^{2})^{2}}$$

20. How is the Rutherford scattering formula recovered from the Yukawa Born result?

    Take the screening parameter $\mu\to0$ with $V_0=\frac{Z_1Z_2e^{2}}{4\pi\varepsilon_0}$, giving $$\frac{d\sigma}{d\Omega}=\left(\frac{2mV_0}{\hbar^{2}q^{2}}\right)^{2}=\left(\frac{Z_1Z_2e^{2}}{16\pi\varepsilon_0 E}\right)^{2}\frac{1}{\sin^{4}(\theta/2)}$$

21. Write the Lippmann-Schwinger integral equation whose iteration generates the Born series.

    $$\psi(\vec r)=\phi(\vec r)+\int G_0(\vec r,\vec r')\,V(\vec r')\,\psi(\vec r')\,d^{3}r'$$ where $\phi$ is the incident wave and $G_0=-\frac{m}{2\pi\hbar^{2}}\frac{e^{ik|\vec r-\vec r'|}}{|\vec r-\vec r'|}$ is the free Green's function. The first Born approximation replaces $\psi\to\phi$ inside the integral.

22. What is the Born-approximation relation between phase shift $\delta_l$ and the potential for small phase shifts?

    $$\delta_l\approx-\frac{2m k}{\hbar^{2}}\int_{0}^{\infty}V(r)\,[\,j_l(kr)\,]^{2}\,r^{2}\,dr$$ valid when $\delta_l\ll1$; $j_l$ is the spherical Bessel function.

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