Write the Fundamental Postulates of Quantum Mechanics.
1. Quantum State Postulate : The quantum state of a physical system is described by a mathematical entity called a wave function (\(\psi\)), also known as a state vector. The wave function contains all the information about the system and can be used to calculate the probabilities of finding the system in different states.
2. Observable and Eigenvalue Postulate : Every physical observable (like position, momentum, energy, etc.) in a quantum system is associated with a mathematical operator (denoted by \(Q\)). When the operator acts on the wave function, it yields the observable's value as an eigenvalue (denoted by \(q\)) and a corresponding eigenstate (\(\psi_q\)). The eigenvalue represents the possible outcome of measuring the observable.
3. Measurement Postulate : When we measure an observable of a quantum system, the system's wave function collapses into one of the eigenstates of the observed observable. The probability of obtaining a specific eigenvalue is given by the square of the absolute value of the coefficient of the corresponding eigenstate in the wave function.
4. Time Evolution Postulate : The time evolution of a quantum system is governed by the Schrödinger equation:
\[i\hbar \frac{{\partial \psi}}{{\partial t}} = H \psi\]
where \(\hbar\) is the reduced Planck's constant, \(H\) is the Hamiltonian operator, and \(\frac{{\partial \psi}}{{\partial t}}\) represents the rate of change of the wave function with time.
5. Quantum Measurement and Superposition : Before measurement, a quantum system can exist in a superposition of multiple states, where each state has an associated probability amplitude. The act of measurement collapses the system into one of these possible states.
6. Uncertainty Principle : There is a fundamental limit to the precision with which certain pairs of complementary observables (e.g., position and momentum) can be simultaneously measured. The more precisely we know one observable, the less precisely we can know the other. Mathematically, the Heisenberg uncertainty principle is given by:
\[\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\]
where \(\Delta x\) is the uncertainty in position and \(\Delta p\) is the uncertainty in momentum.
These postulates form the foundation of quantum mechanics and help us understand the strange and counterintuitive behavior of particles at the quantum level. They have been tested and validated through numerous experiments and have become the basis for modern technologies like quantum computing and quantum cryptography.
Also Read : Write the Zeroth Law of Thermodynamics.
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