GATE Physics 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 GATE Physics, not a summary of it.

1. 1

   Postulates of Quantum Mechanics

   4 topics

   • Uncertainty Principle
   • Schrodinger Equation
   • Dirac Bra-Ket Notation
   • Linear Vectors and Operators in Hilbert Space
2. 2

   One Dimensional Potentials

   5 topics

   • Step Potential
   • Finite Rectangular Well
   • Tunneling from a Potential Barrier
   • Particle in a Box
   • Harmonic Oscillator
3. 3

   Two and Three Dimensional Systems

   4 topics

   • Concept of Degeneracy
   • Hydrogen Atom
   • Angular Momentum and Spin
   • Addition of Angular Momenta
4. 4

   Variational Method and WKB Approximation

   1 topic

   • Time Independent Perturbation Theory
5. 5

   Elementary Scattering Theory

   1 topic

   • Born Approximation
6. 6

   Symmetries in Quantum Mechanical Systems

   overview

   Examined as a single unit within Quantum Mechanics — no further topic split in the official outline.

Quantum Mechanics flashcards for GATE Physics

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

1. State the Heisenberg uncertainty principle for position and momentum, including the limiting value.

   The product of the uncertainties in position and momentum satisfies $$\Delta x \,\Delta p_{x} \geq \frac{\hbar}{2},$$ where $\hbar = \frac{h}{2\pi}$. It is impossible to simultaneously know both $x$ and $p_{x}$ with arbitrary precision.

2. Write the energy-time uncertainty relation and explain its physical meaning.

   $$\Delta E \,\Delta t \geq \frac{\hbar}{2}.$$ A state that exists for a finite time $\Delta t$ has an energy spread $\Delta E$; only states living infinitely long have perfectly sharp energy.

3. What is the general uncertainty relation between two observables $\hat{A}$ and $\hat{B}$?

   $$\Delta A\,\Delta B \geq \frac{1}{2}\left| \langle [\hat{A},\hat{B}] \rangle \right|,$$ where $[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$. Observables with a nonzero commutator cannot be measured simultaneously to arbitrary precision.

4. Write the time-dependent Schrodinger equation in one dimension.

   $$i\hbar \frac{\partial \Psi(x,t)}{\partial t} = -\frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi(x,t)}{\partial x^{2}} + V(x,t)\,\Psi(x,t).$$

5. Write the time-independent Schrodinger equation in three dimensions.

   $$-\frac{\hbar^{2}}{2m}\nabla^{2}\psi(\vec{r}) + V(\vec{r})\,\psi(\vec{r}) = E\,\psi(\vec{r}),$$ obtained by separation of variables with $\Psi(\vec{r},t) = \psi(\vec{r})\,e^{-iEt/\hbar}$.

6. What is the probability current density $j(x,t)$ in one dimension, and what equation does it satisfy with the probability density?

   $$j(x,t) = \frac{\hbar}{2mi}\left( \Psi^{*}\frac{\partial \Psi}{\partial x} - \Psi\frac{\partial \Psi^{*}}{\partial x} \right).$$ It satisfies the continuity equation $\frac{\partial \rho}{\partial t} + \frac{\partial j}{\partial x} = 0$ with $\rho = |\Psi|^{2}$.

7. State the normalization condition and the standard boundary/continuity conditions on a physically acceptable wavefunction.

   Normalization: $\int_{-\infty}^{\infty} |\psi|^{2}\,dx = 1$. The wavefunction $\psi$ must be single-valued, finite, and continuous everywhere; $\frac{d\psi}{dx}$ must also be continuous wherever $V$ is finite.

8. In Dirac bra-ket notation, what are a ket, a bra, and how is the inner product written?

   A ket $|\psi\rangle$ is a state vector; the bra $\langle\psi|$ is its dual (Hermitian conjugate). The inner product is $\langle\phi|\psi\rangle$, a complex number with $\langle\phi|\psi\rangle = \langle\psi|\phi\rangle^{*}$.

9. Write the completeness (closure) relation and the orthonormality condition for a discrete orthonormal basis $\{|n\rangle\}$.

   Orthonormality: $\langle m|n\rangle = \delta_{mn}$. Completeness: $$\sum_{n} |n\rangle\langle n| = \hat{I},$$ where $\hat{I}$ is the identity operator.

10. In bra-ket notation, how is the expectation value of an operator $\hat{A}$ in a normalized state $|\psi\rangle$ written, and what is the projection operator onto $|n\rangle$?

    Expectation value: $\langle \hat{A} \rangle = \langle\psi|\hat{A}|\psi\rangle$. The projection operator onto $|n\rangle$ is $\hat{P}_{n} = |n\rangle\langle n|$, which satisfies $\hat{P}_{n}^{2} = \hat{P}_{n}$.

11. Define a Hermitian operator and state two key properties of its eigenvalues and eigenvectors.

    An operator is Hermitian if $\hat{A} = \hat{A}^{\dagger}$, i.e. $\langle\phi|\hat{A}\psi\rangle = \langle\hat{A}\phi|\psi\rangle$. Its eigenvalues are real, and eigenvectors belonging to distinct eigenvalues are orthogonal. All physical observables are represented by Hermitian operators.

12. Define a Hilbert space in the context of quantum mechanics.

    A Hilbert space is a complete, complex inner-product vector space. Quantum states are vectors in this space, observables are linear (Hermitian) operators on it, and completeness guarantees convergence of vector expansions in a basis.

13. Define a unitary operator and state its defining property and physical role.

    A unitary operator satisfies $\hat{U}^{\dagger}\hat{U} = \hat{U}\hat{U}^{\dagger} = \hat{I}$, so $\hat{U}^{-1} = \hat{U}^{\dagger}$. It preserves inner products and norms, and represents symmetry transformations and time evolution, e.g. $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$.

14. What is the canonical commutation relation between position and momentum operators, and the momentum operator in position space?

    $$[\hat{x},\hat{p}] = i\hbar.$$ In the position representation, $\hat{p} = -i\hbar\frac{\partial}{\partial x}$ and $\hat{x} = x$.

15. For a step potential of height $V_{0}$ with incident energy $E > V_{0}$, qualitatively what happens, and is there reflection?

    The particle is partly transmitted and partly reflected. Unlike classical physics, there is a nonzero reflection probability even though $E > V_{0}$. The transmitted wave has reduced wave number $k_{2} = \sqrt{2m(E-V_{0})}/\hbar < k_{1}$.

16. For a step potential of height $V_{0}$ with $E < V_{0}$, what is the reflection coefficient and the nature of the wavefunction inside the step?

    The reflection coefficient is $R = 1$ (total reflection). Inside the step the wavefunction is evanescent (exponentially decaying), $\psi \sim e^{-\kappa x}$ with $\kappa = \frac{\sqrt{2m(V_{0}-E)}}{\hbar}$, so the penetration probability is nonzero but no net transmission occurs.

17. Write the reflection coefficient $R$ for a step potential with $E > V_{0}$ in terms of wave numbers $k_{1}$ and $k_{2}$.

    $$R = \left( \frac{k_{1}-k_{2}}{k_{1}+k_{2}} \right)^{2},$$ with $k_{1} = \frac{\sqrt{2mE}}{\hbar}$ and $k_{2} = \frac{\sqrt{2m(E-V_{0})}}{\hbar}$. Transmission satisfies $T = 1 - R$.

18. For a finite rectangular (square) potential well, how do the number and nature of bound states compare with the infinite well?

    A finite well supports only a finite number of bound states (at least one always exists in 1D). The wavefunctions leak (decay exponentially) into the classically forbidden regions outside the well, and bound-state energies are lower than the corresponding infinite-well levels.

19. For a finite square well, what type of equation determines the bound-state energies and why must it be solved graphically or numerically?

    Matching $\psi$ and $\psi'$ at the walls yields transcendental equations such as $k\tan(ka) = \kappa$ (even states) and $k\cot(ka) = -\kappa$ (odd states), where $k=\sqrt{2mE}/\hbar$ inside and $\kappa=\sqrt{2m(V_0-E)}/\hbar$ outside. These have no closed-form solution and must be solved graphically/numerically.

20. What is quantum tunneling through a potential barrier?

    Tunneling is the quantum phenomenon where a particle with energy $E < V_{0}$ has a nonzero probability of transmitting through a barrier, even though it is classically forbidden. Inside the barrier the wavefunction decays exponentially but remains finite on the far side.

21. Give the approximate transmission coefficient for tunneling through a wide/high rectangular barrier of width $L$ and height $V_{0}$ (with $E < V_{0}$).

    $$T \approx e^{-2\kappa L}, \qquad \kappa = \frac{\sqrt{2m(V_{0}-E)}}{\hbar}.$$ Transmission falls off exponentially with barrier width and with the square root of $(V_{0}-E)$.

22. State the WKB (Gamow) tunneling formula for transmission through an arbitrary smooth barrier.

    $$T \approx \exp\!\left( -\frac{2}{\hbar}\int_{x_{1}}^{x_{2}} \sqrt{2m\,[V(x)-E]}\;dx \right),$$ where $x_{1}$ and $x_{2}$ are the classical turning points where $V(x)=E$.

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