atomic structure

Question

how solutions of Schrodinger's equation contribute towards electron density curve of hydrogen atom?

SAGAR SINGH - IIT DELHI · Accepted Answer

Dear anurag,

The solution of the Schrodinger equation for the hydrogen atom is a formidable mathematical problem, but is of such fundamental importance that it will be treated in outline here. The solution is managed by separating the variables so that the wavefunction is represented by the product:

The separation leads to three equations for the three spatial variables, and their solutions give rise to three quantum numbers associated with the hydrogen energy levels.

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General quantum system

For a general quantum system:^[2]

$i\hbar\frac{\partial}{\partial t} \Psi = \hat H \Psi$

where

$Ψ$ is the wave function; the probability amplitude for different configurations of the system at different times,
$i \hbar \frac {\partial}{\partial t}$ is the energy operator ( $i$ is the imaginary unit and $\hbar$ is the reduced Planck constant),
$\hat H$ is the Hamiltonian operator.

[edit] Single particle in a potential

For a single particle with potential energy V, the Schrödinger equation takes the form:^[3]

$i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) = -\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t) + V(\mathbf{r})\Psi(\mathbf{r},\,t)$

where

$-\frac{\hbar^2}{2m}\nabla^2$ is the kinetic energy operator, where m is the mass of the particle.
$\nabla^2$ is the Laplace operator. In three dimensions, the Laplace operator is $\frac{\partial^2}{{\partial x}^2} + \frac{\partial^2}{{\partial y}^2} + \frac{\partial^2}{{\partial z}^2}$ , where x, y, and z are the Cartesian coordinates of space.
$V\left(\mathbf{r}\right)$ is the time-independent potential energy at the position r.
$\Psi(\mathbf{r},\,t)$ is the probability amplitude for the particle to be found at position r at time

Physical Chemistry> atomic structure...

how solutions of Schrodinger's equation contribute towards electron density curve of hydrogen atom?

Anurag Chakraborty , 15 Years ago

SAGAR SINGH - IIT DELHI

General quantum system

[edit] Single particle in a potential

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Physical Chemistry> atomic structure...

how solutions of Schrodinger's equation contribute towards electron density curve of hydrogen atom?

Anurag Chakraborty , 15 Years ago

SAGAR SINGH - IIT DELHI

General quantum system

[edit] Single particle in a potential

Srinivas Rao

LIVE ONLINE CLASSES

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