The Schrödinger Equation | ChemTalk – keiseronlineuniversity.com

Chemistry

The Schrödinger Equation | ChemTalk

keiseronlineuniversity.com

8 September 2023

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Core Ideas

On this tutorial you’ll study what the Schrödinger Equation is, and the way it may be used to know the conduct of quantum techniques.

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What’s the Schrödinger Equation?

begin{gather*} Hat{H} text{ is the Hamiltonian} psi text{ is a wavefunction} E text{ is the energy of the wavefunction} end{gather*}

The Schrödinger equation is the elemental quantum mechanical equation which relates the wavefunction of a system to its vitality. It’s a linear partial differential equation which makes use of the operator often called the Hamiltonian to find out the vitality of a system. The Hamiltonian has one time period for the kinetic vitality and one time period for the potential vitality of the system. The Hamiltonian is completely different for every system it’s used to review. The Schrödinger Equation is extraordinarily helpful in that it may be solved for the energies and wavefunctions of a quantum system and serves as the inspiration for our understanding of atoms and molecules.

The One-Dimensional Schrödinger Equation

One of many easiest methods to make use of the Schrödinger equation is to find out the vitality of an electron in a one-dimensional field. The electron has wavefunctions which are standing waves throughout the field the electron is confined to.

$begin{gather*} { -frac{hbar^2}{2m_e}frac{d^2psi}{dx^2}+Vpsi=Epsi }end{gather*}$

V is a operate describing the potential of the system. On this case, the potential is zero throughout the field, and infinite exterior the field, guaranteeing that the electron will stay within the field. The differential equation is solved to yield a wavefunction and an vitality. This differential equation additionally has boundary circumstances for all sides of the field.

$begin{gather*} text{On one side:} lim_{x to 0} psileft(xright)=0 text{and on the other side:} lim_{x to L} psileft(xright)=0 end{gather*}$

The one-dimensional Schrödinger equation isn’t utilized in observe to precisely decide the vitality of a system. The mannequin’s simplicity is useful in understanding the vitality of straightforward digital techniques, and the way variations in vitality ranges are derived from nodal variations. Nevertheless, the appliance of this Hamiltonian to even the most straightforward atoms renders it pretty ineffective because of its inaccuracy.

Fixing the Schrödinger Equation

The final answer to the Schrödinger Equation for the particle in a field is:

Contemplating the aforementioned boundary circumstances, we are able to simplify the answer additional to incorporate solely the sine part.

$begin{gather*} { psileft(xright)=Asin(kx)} {k=frac{npi}{L}}end{gather*}$

The Quantum Harmonic Oscillator Schrödinger Equation

Whereas the one-dimensional Schrödinger equation can account for the kinetic and potential vitality of an atom considerably persistently, it doesn’t account for the oscillatory conduct of quantum techniques derived from their varied wavefunctions. To account for the potential variations caused by equilibria factors we use the hamiltonian proven beneath.

$begin{gather*} {Hat{H}=frac{{Hat{p}}^2}{2m} + frac{1}{2}k{{Hat{x}}^2} = frac{{Hat{p}}^2}{2m} + frac{1}{2}m{omega^2}{{Hat{x}}^2} }end{gather*}$

$begin{gather*} Hat{p}=-i{hbar}frac{d}{dx} text{ is the momentum operator} m text{ is the mass} k text{ is the force constant} Hat{x} text{ is the position operator} omega text{ is the angular frequency} end{gather*}$

This Hamiltonian ends in equally spaced vitality ranges, in contrast to the one dimensional “particle in a field” mannequin. This mannequin is commonly used to described the vibrational modes of molecules. The harmonic oscillator potential describes the spring-like nature of the chemical bonds between the atoms.

The Three-Dimensional Schrödinger Equation

Whereas the one-dimensional Schrödinger equation could also be easy to resolve, many techniques we want to research exist in three dimensions. The three dimensional Schrödinger equation could also be written in Cartesian coordinates or spherical coordinates, relying on which inserts the system greatest.

$begin{gather*} { -{frac{h^2}{2{m_e}}}left({frac{{partial^2}}{partial{x^2}}}+{frac{{partial^2}}{partial{y^2}}}+{frac{{partial^2}}{partial{z^2}}}right)psi+Vpsi=Epsi }end{gather*}$

Nevertheless, this Hamiltonian is commonly simplified utilizing an emblem referred to as “del”. Squaring this operator means making use of it twice, or taking the second partial spinoff with respect to every coordinate.

$begin{gather*} { {nabla^2} ={frac{{partial^2}}{partial{x^2}}}+{frac{{partial^2}}{partial{y^2}}}+{frac{{partial^2}}{partial{z^2}}}}end{gather*}$

Utilizing Spherical Polar Coordinates

Electrons in atoms are most naturally described utilizing spherical coordinates, the place the nucleus is fastened on the origin and the place of an electron is described utilizing the radial place r, the polar angle θ, and the azimuthal angle φ.

$begin{gather*} { {nabla^2} =frac{1}{r^2} {frac{{partial}}{partial{r}}} left({r^2}{frac{{partial}}{partial{r}}}right) + frac{1}{{r^2}sin(theta)}{frac{{partial}}{partial{theta}}} left({sin(theta)}{frac{{partial}}{partial{theta}}}right) + frac{1}{{r^2}{sin(theta)}^2}{frac{{partial}^2}{partial{theta}^2}} }end{gather*}$

Eigenfunctions and Eigenvalues

Fixing to the Schrödinger equation yields eigenvalues and eigenfunctions. These eigenfunctions are the wavefunctions of the system, and each has a definite eigenvalue, representing a discreet vitality the system can have. A classical interpretation of the system would permit the vitality to fluctuate repeatedly, nevertheless quite a few experiments help the quantum mannequin, the place the system might have solely a discreet set of energies. Different quantum operators moreover the Hamiltonian can have eigenvalues and eigenfunctions too, that means that vitality is just not the one amount that turns into quantized on the molecular and atomic stage. One other instance is angular momentum, which leads to completely different shapes for s, p, d, and different atomic orbitals.

Potential Power

The V time period within the Schrödinger equation represents potential vitality. It varies relying on the system being studied. The bond in a diatomic molecule could also be described utilizing a harmonic potential vitality operate, whereas extra complicated molecules have a number of phrases, accounting for electron-electron repulsion in addition to electron-nucleus attraction.

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