Question

State 1st , 2nd , 3rd and 4th Postulates of Quantum mechanics with appropriate
examples

Answer 1

Answer:

In this section, we will present six postulates of quantum mechanics. Again, we follow the presentation of McQuarrie [1], with the exception of postulate 6, which McQuarrie does not include. A few of the postulates have already been discussed in section 3.

   Postulate 1. The state of a quantum mechanical system is completely specified by a function $\Psi({\bf r}, t)$ 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 $\Psi^{*}({\bf r}, t) \Psi({\bf r}, t) d\tau$ is the probability that the particle lies in the volume element $d\tau$ located at ${\bf r}$ at time $t$.

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

\begin{displaymath} \int_{-\infty}^{\infty} \Psi^{*}({\bf r}, t) \Psi({\bf r}, t) d\tau = 1 \end{displaymath}  (110)

It is customary to also normalize many-particle wavefunctions to 1.2 The wavefunction must also be single-valued, continuous, and finite.

   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 $\hat{A}$ is real, then $\hat{A}$ must be a Hermitian operator. Some common operators occuring in quantum mechanics are collected in Table 1.

Explanation:

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