Postulates of Quantum Mechanics

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Here are the postulates of quantum mechanics,

1. State postulate: 

All time independent states of a quantum system are described by a normalized vector in a complex Hilbert space. The vector represents the system's wavefunction. That function is single valued, continuous and finite. 

2. Probability of describing quantum mechanical system:

The probability of describing a quantum state is determined by the state vector of the system. 

The state vector can be written as a linear combination of basis states, where each basis state represents a possible outcome of a measurement. The probability of a particular outcome (measurement result) is given by the squared magnitude of the coefficient corresponding to that basis state.

For example, if a quantum state can be represented as a linear combination of basis states |0⟩ and |1⟩ with coefficients α and β respectively, then the probability of measuring outcome 0 would be |α|^2, and the probability of measuring outcome 1 would be |β|^2.

It is important to note that the probabilities of all possible outcomes sum up to 1, as the state vector must be normalized. Therefore, if there are n possible outcomes, the sum of the squared magnitudes of the coefficients must be equal to 1: |α|^2 + |β|^2 + ... + |γ|^2 = 1.

Thus, the probability of describing quantum states is determined by the squared magnitudes of the coefficients of the basis states in the state vector.

3. Eigen value postulate: 

For measurement of a physical observable in a quantum system, an operator operates over the wave function of the system. The outcome of the measurement is one of the eigenvalues of the observable. (More on that here.) 

4. The Operators: 

For every physical property in a quantum mechanical system, there’s a particular operator whose nature depends on classical expression of the property in consideration. (More about Operators is covered here.)

5. Schrodinger Wave Equation postulate: 

The time evolution of the system's wavefunction is governed by the Schrödinger equation, which is a linear, first-order, partial differential equation. It describes how the wavefunction changes over time. (More about schrodinger Wave Equation is covered here.)

4. Superposition postulate: 

The superposition principle represents physically admissible states and is valid for all Wave functions. The state vector of a quantum system can exist in a superposition of multiple eigenstates. When measured, the system's wavefunction collapses into one of the eigenstates. (More on that here.)

6. Expectation value postulate: 

If the value of physical property that was obtained by operating a Wave function with an operator is variable or non-eigen, there comes the role of expectation value of the operator.

Expectation value is the value obtained by averaging it over the whole space. More about expectation value here.