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**Postulates of Quantum Mechanics**

Here are the Six Postulates of Quantum Mechanics. Postulate 2 contains various Operators use in Quantum Mechanics.

**Postulate 1.** The state of a quantum mechanical system is completely specified by a function **ψ(r, t) **that depends on the coordinates of the particle(s) and on time. This function, called the wave function or state function, has the important property that **ψ(r, t) ****ψ(r, t) dτ **is the probability that the particle lies in the volume element ** dτ **located at

The wavefunction must satisfy certain mathematical conditions because of this probabilistic interpretation. For the case of a single particle, the probability of finding it somewhere is 1, so that we have the normalization condition

(110)

It is customary to also normalize many-particle wavefunctions to 1.

**Postulate 2.** To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics.

This postulate comes about because of the considerations raised in section 3.1.5: if we require that the expectation value of an operator Â is real, then Â must be a Hermitian operator. Some common operators occuring in quantum mechanics are collected in Table 1.**Table 1:** Physical observables and their corresponding quantum operators (single particle)

**Postulate 3.** In any measurement of the observable associated with operator Â, the only values that will ever be observed are the eigenvalues __a__, which satisfy the eigenvalue equation

ÂΨ = aΨ (111)

This postulate captures the central point of quantum mechanics--the values of dynamical variables can be quantized (although it is still possible to have a continuum of eigenvalues in the case of unbound states). If the system is in an eigenstate of **Â** with eigenvalue __ a__, then any measurement of the quantity

Although measurements must always yield an eigenvalue, the state does not have to be an eigenstate of

(112)

where

An important second half of the third postulate is that, after measurement of

**Postulate 4.** If a system is in a state described by a normalized wave function **Ψ**, then the average value of the observable corresponding to **Â** is given by

(113)

**Postulate 5. **The wavefunction or state function of a system evolves in time according to the time-dependent Schrödinger equation

(114)

The central equation of quantum mechanics must be accepted as a postulate, as discussed in section 2.2.

**Postulate 6. **The total wavefunction must be antisymmetric with respect to the interchange of all coordinates of one fermion with those of another. Electronic spin must be included in this set of coordinates.

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