Postulate of wave mechanics or Quantum Mechanics

Postulate of Wave or Quantum Mechanics (or Operator formalism in Quantum mechanics) →

The formulation of mathematical equations of quantum mechanics is based on the linear operator. This operator formulation of quantum mechanics is known as postulates of quantum mechanics. These postulates are given below:-

  1. For a system consisting of particles moving in a field of a conservative force, there is an associated complex wave function $\psi(x, y, z, t)$ where $x$,$y$,$z$ space coordinates, and $t$ is the time. This wave function enables us to obtain a description of the behavior of the system, consistent with the principle of uncertainty.

  2. There is an operator with every observable dynamical quantity. The operator corresponding to the pertinent dynamical quantities is:-

    Dynamical Variable Symbol Quantum Mechanical Operator
    Position $x$
    Momentum $P_{x}$
    Generalised Form $\overrightarrow{P}$
    $\frac{\hbar}{i}\frac{\partial}{\partial x}$
    $\frac{\hbar}{i}\frac{\partial}{\partial y}$
    $\frac{\hbar}{i}\frac{\partial}{\partial z}$
    Generalised Form $\frac{\hbar}{i}\overrightarrow{\nabla}$
    Total Energy $E$ $i\hbar \frac{\partial}{\partial t}$
    Total Energy $E$ $-\frac{\hbar^{2}}{2m} \nabla^{2}+V(x)$
    This is also known as Hamiltonian Operator $H$.
    Kinetic Energy $K$ $-\frac{\hbar^{2}}{2m} \nabla^{2}$
    Potential Energy $V(x,y,z)$ $V(x,y,z)$

    All the operators have eigen functions and eigen values.

  3. The wave function $\psi(x,y,z,t)$ and its partial derivatives $\frac{\partial \psi}{\partial x}$, $\frac{\partial \psi}{\partial y}$, $\frac{\partial \psi}{\partial z}$ must be finite, continuous and single-valued for all values of $x$, $y$, $z$ and $t$

  4. The product of $\psi(x,y,z,t)$ and $\psi^{*}(x,y,z,t)$ is always a real quantity. The product is called the probability density and $\psi\psi^{*} d\tau $ is interpreted as a probability that the particle will be found in volume element $d\tau$ at $x$,$y$,$z$ and time $t$. Since the total probability of finding particles somewhere in the entire space must be equal to 1.

    $\int_{-\infty}^{+\infty} \psi \: \psi^{*}\: d\tau = 1$

    The integral is taken overall space.

  5. The average or expectation value of an observable quantity $\alpha$ with which an operator $\hat{\alpha}$ is associated is defined by

  6. $\left< \alpha \right> = \int_{-\infty}^{+\infty} \psi^{*} \hat{\alpha} \psi d\tau$

    The integral being taken overall space.

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