Introduction
Classical mechanics fails to describe the dynamics of atomic and subatomic particles. Quantum Mechanics replaces deterministic trajectories with state vectors and probabilities. The operational rules of this mathematical framework are summarized by five core postulates.
Postulate 1: The State of a System
At any time \(t\), the state of a physical system is represented by a vector \(|\psi(t)\rangle\) in a complex vector space equipped with an inner product (a **Hilbert space**). The wave function in coordinate space is the projection: \[\psi(x,t) = \langle x | \psi(t) \rangle\] The state vector contains all possible physical information about the system. For a single particle, the probability of finding the particle between \(x\) and \(x+dx\) is \(|\psi(x)|^2 dx\), requiring the normalization condition: \[\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1\]
Postulate 2: Observables and Operators
To every physically measurable quantity (observable) like position, momentum, energy, or angular momentum, there corresponds a **linear Hermitian operator** acting on the Hilbert space.
An operator \(\hat{A}\) is Hermitian if:
\[\langle \phi | \hat{A} \psi \rangle = \langle \hat{A} \phi | \psi \rangle\]
Hermitian operators have two crucial properties:
1. Their eigenvalues are strictly real numbers (matching physical measurements).
2. Eigenvectors corresponding to distinct eigenvalues are orthogonal.
Postulate 3 & 4: Measurement and Born's Rule
When measuring an observable \(A\), the only values we can obtain are the eigenvalues \(a_n\) of its operator \(\hat{A}\), solved from: \[\hat{A} |\phi_n\rangle = a_n |\phi_n\rangle\] If the system is in a normalized state \(|\psi\rangle\), the probability of measuring \(a_n\) is: \[P(a_n) = |\langle \phi_n | \psi \rangle|^2\] If the eigenvalue spectrum is continuous (like position or momentum), the probability of finding the measurement in range \([a, b]\) is: \[P(a \le x \le b) = \int_{a}^{b} |\psi(x)|^2 dx\] Upon measurement, the state of the system immediately collapses into the corresponding eigenvector \(|\phi_n\rangle\).
Postulate 5: Time Evolution
Between measurements, the state vector evolves continuously in time according to the **Time-Dependent Schrödinger Equation**: \[i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle\] Where \(\hat{H}\) is the Hamiltonian operator (representing the total energy of the system), and \(\hbar = \frac{h}{2\pi}\) is the reduced Planck constant.
Solved CSIR-NET & GATE Physics Questions
Question 1 (CSIR-NET Physical Sciences)
A quantum system is in a state \(|\psi\rangle = \frac{1}{\sqrt{3}} |\phi_1\rangle + i\sqrt{\frac{2}{3}} |\phi_2\rangle\), where \(|\phi_1\rangle\) and \(|\phi_2\rangle\) are orthonormal energy eigenstates with energy eigenvalues \(E_1\) and \(E_2\) respectively. Find the expectation value of energy \(\langle E \rangle\).
Detailed Solution:
- Calculate the probabilities (\(P_n\)) of measuring each energy state:
• \(P(E_1) = |c_1|^2 = \left|\frac{1}{\sqrt{3}}\right|^2 = \frac{1}{3}\)
• \(P(E_2) = |c_2|^2 = \left|i\sqrt{\frac{2}{3}}\right|^2 = \frac{2}{3}\)
Check normalization: \(P_1 + P_2 = \frac{1}{3} + \frac{2}{3} = 1\) (normalized). - Expectation value is the weighted average of eigenvalues:
⟨ E ⟩ = P(E_1)E_1 + P(E_2)E_2
- Substitute values:
\(\langle E \rangle = \frac{1}{3} E_1 + \frac{2}{3} E_2\)
Answer: \(\langle E \rangle = \frac{E_1 + 2E_2}{3}\).
Question 2 (GATE Physics)
Why must physical observables be represented by Hermitian operators? Select the correct mathematical proof.
Detailed Solution:
For an operator \(\hat{A}\) representing a physical measurement, its eigenvalues \(a\) must be real, meaning \(a = a^*\).
Let \(\hat{A} |\phi\rangle = a |\phi\rangle\). Taking the inner product with \(\langle \phi |\): \[\langle \phi | \hat{A} | \phi \rangle = a \langle \phi | \phi \rangle\]
If \(\hat{A}\) is Hermitian, we also know: \[\langle \phi | \hat{A} | \phi \rangle = \langle \hat{A} \phi | \phi \rangle = a^* \langle \phi | \phi \rangle\]
Equating the two expressions: \[(a - a^*) \langle \phi | \phi \rangle = 0\] Since \(|\phi\rangle\) is a physical state, \(\langle \phi | \phi \rangle \ne 0\). Therefore: \[a = a^*\]
Answer: Hermitian operators guarantee real eigenvalues, matching observable experimental measurements.
Dr. Saju K John
Senior Physics Faculty | PhD (NIT Calicut) | GATE AIR-52Dr. Saju K John is a PhD graduate from NIT Calicut. He cleared CSIR-NET and secured **GATE AIR 52** in Physics. He directs the Advanced Thermal and Quantum Physics syllabus modules at Benzil.