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

Question

SAGAR SINGH - IIT DELHI · 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

Srinivas Rao · Answer

schrodinger equation is a differential equation written in terms of electron wave function. So, the equation is a store house of information about the revolving electron. As squre of amplitude speaks about the intensity of a wave, square of si gives probability of finding the electron.

4(pi) r²gives area of sphere, 4(pi) r²dr gives a volume segment.

4(pi) r²dr (si) square gives radial probability distribution in a volume segment with out the consideration of any direction, that is nothing but electron cloud density,

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

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

2 Answers

General quantum system

[edit] Single particle in a potential

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

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

2 Answers

General quantum system

[edit] Single particle in a potential

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