Schrodinger Wave Equation

Time - Independent Schroodinger Equation


Schrodinger’s Equation refers to a fundamental equation of quantum physics. In classical physics, it is parallel to Newton’s Laws of Motion, which helps you to calculate the future position and momentum of the object if you know the present position and momentum of an object. Although parallel, Schrodinger’s Equation is not deterministic as Newton’s laws. Newton's laws are deterministic because by using the given knowledge of the initial position and the measurements of the forces acting on the object, one can tell how the forces will interact, and therefore, where the object is going to be in the upcoming point of time. In 1925, Schrodinger and Heisenberg independently synthesized the representations of quantum mechanics that successfully describe physical phenomena at the microscopic level of nuclei, molecules, and atoms. Here, in the following article, we will discuss Schrodinger’s equation in deep.


Schrodinger’s Equation doesn't tell the position of the subatomic particles at any future point in time. It will tell only the possible positions and probabilities of being in those possible positions. For instance, if you use a laser to shoot some photons towards a photographic plate, this equation can help you calculate the overall pattern of pixels that will form on the plate, but not the position of pixels the particular photon would light up. Hence, we can say that the Schrodinger’s Equation is deterministic but only at the statistical level, not at the individual particle level.


Another fact about Schrodinger’s Equation is that it is open to considerable interpretation and the nature of the physical reality that describes it.


The Schrodinger Equation comes up as a mathematical expression. It describes the transformation of the physical quantity overtime, where the quantum effects like a wave-particle duality. The equation has two forms, the time-independent Schrodinger equation and the time-dependent Schrodinger equation. 


The Time-Dependent Schrodinger Equation

De Broglie relation cannot be derived by using elementary methods although we are able to derive this equation starting from the classical wave equation. We can show that the time-dependent equation, if not derivable, is at least reasonable, and the arguments are rather involved.


Schrodinger Wave Equation Derivation (Time-Dependent)

The single-particle time-dependent Schrodinger equation is,

\begin{displaymath}
i \hbar \frac{\partial \psi({\bf r},t)}{\partial t} =
- \f...
...r^2}{2m} \nabla^2 \psi({\bf r},t) + V({\bf r}) \psi({\bf r},t)
\end{displaymath}

Where

V represents the potential energy and is assumed to be a real function

Now, if we write the wave function as a product of temporal and spatial terms, then the equation will become,


\begin{displaymath}
\psi({\bf r}) i \hbar \frac{df(t)}{dt} = f(t) \left[
- \frac{\hbar^2}{2m} \nabla^2 + V({\bf r}) \right] \psi({\bf r})
\end{displaymath}


or

\begin{displaymath}
\frac{i \hbar}{f(t)} \frac{df}{dt} = \frac{1}{\psi({\bf r})}...
...frac{\hbar^2}{2m} \nabla^2 + V({\bf r}) \right] \psi({\bf r})
\end{displaymath}


Since the right-hand side is a function of r only and the left-hand side is of t only, the two sides should equal a constant. In cases where we designate the constant E, the two ordinary differential equation, namely

\begin{displaymath}
\frac{1}{f(t)} \frac{df(t)}{dt} = - \frac{i E}{\hbar}
\end{displaymath}

and


\begin{displaymath}
-\frac{\hbar^2}{2m} \nabla^2\psi({\bf r}) + V({\bf r}) \psi({\bf r}) =
E \psi({\bf r})
\end{displaymath}



Here, the former equation is solved to get, \begin{displaymath}
f(t) = e^{-iEt / \hbar}
\end{displaymath}

However, the latter equation is the time-independent Schrödinger equation

Considering a complex plane wave:

Derivation Of Schrodinger Wave Equation

Now the Hamiltonian of a system is

Derivation Of Schrodinger Wave Equation

Where 

T is the kinetic energy and V is the potential energy. As we know that H is the total energy, we can rewrite the equation as:

Derivation Of Schrodinger Wave Equation

Now, by taking the derivatives, we get

Derivation Of Schrodinger Wave Equation

All of us know that,

Derivation Of Schrodinger Wave Equation

Where 

‘λ’ is the wavelength 

‘k’ is the wave number.

Now, as we have

Derivation Of Schrodinger Wave Equation

Hence,

Derivation Of Schrodinger Wave Equation

Here,by multiplying the Hamiltonian to Ψ (x, t), we get,

Derivation Of Schrodinger Wave Equation

The above expression can also be written as:

Derivation Of Schrodinger Wave Equation

As the energy of a matter wave is

Derivation Of Schrodinger Wave Equation

So, we can say that

Derivation Of Schrodinger Wave Equation

Now, by combining the parts, we can get the Schrodinger Wave Equation.

Derivation Of Schrodinger Wave Equation

So, this was the derivation of the Schrodinger Wave Equation (time-dependent)


Schrodinger Wave Equation Derivation (Time-Dependent)

How to extract the knowledge about momenta from Ψ(qj,t) is treated below, where the structure of quantum mechanics, the use of operators and wave functions to make predictions and interpretations about experimental measurements, and the origin of 'uncertainty relations' such as the well-known Heisenberg uncertainty condition dealing with measurements of coordinates and momenta are also treated.


Before moving deeper to understand what quantum mechanics actually 'means,' it is essential to learn how the wave functions ΨΨ are found by applying the basic equation of quantum mechanics, the Schrodinger equation, to a few exactly soluble model problems. Being aware of the solutions to these easy yet chemically relevant models will help you in being familiar with more details of the structure of quantum mechanics because these model cases can be used as 'concrete examples.'

The Schrodinger equation is a differential equation based on all the spatial coordinates necessary to describe the system at hand and time (thirty-nine for the H2O example cited above). It is usually written as

HΨ=iℏ∂Ψ∂t(1.3.1)(1.3.1)HΨ=iℏ∂Ψ∂t

Where

Ψ(qjΨ(qj,t) is the unknown wave function 


H H is the operator corresponding to the total energy physical property of the system. This operator is called the Hamiltonian and is formed by first writing the classical mechanical expression for the total energy (potential + kinetic) in Cartesian coordinates and momenta and then replacing all the classical momenta 'pj' by the quantum mechanical operators pj=−iℏ∂∂qjpj=−iℏ∂∂qj. For H2O example mentioned above, the classical/mechanical energy of all the thirteen particles is

E=∑i(p2i2me+12∑je2ri,j−∑aZae2ri,a)+∑a(−ℏ22ma∂2∂q2a+12∑bZaZbe2ra,b)(1.3.2)(1.3.2)E=∑i(pi22me+12∑je2ri,j−∑aZae2ri,a)+∑a(−ℏ22ma∂2∂qa2+12∑bZaZbe2ra,b)


Where 

the indices i and j label the ten electrons whose thirty cartesian coordinates are {qii} 

the a and b label the three nuclei whose charges are represented by {Zaa}, and the nine cartesian coordinates are {qaa}. The electron and nuclear masses are denoted as me and {maa}, respectively.The corresponding Hamiltonian operator is

H=∑i(−(ℏ22me)∂2∂q2i+12∑je2ri,j−∑aZae2ri,a)+∑a(−(ℏ22ma)∂2∂q2a+12∑bZaZbe2ra,b).H=∑i(−(ℏ22me)∂2∂qi2+12∑je2ri,j−∑aZae2ri,a)+∑a(−(ℏ22ma)∂2∂qa2+12∑bZaZbe2ra,b).

Note that H is a second-order differential operator in the list of the thirty-nine Cartesian coordinates describing the positions of the three nuclei and ten electrons. The fact, which makes it a second-order operator, is that the quantum mechanical operator for every momentum p = iℏ∂∂qiℏ∂∂q is of the first order and momenta appear in the kinetic energy as p2jpj2 and p2apa2.

The Schrodinger equation for the H2O then reads

∑i[−(ℏ22me)∂2∂q2i+12∑je2ri,j−∑aZae2ri,a]Ψ+∑[−(ℏ22ma)∂2∂q2a+12∑bZaZbe2ra,b]Ψ∑i[−(ℏ22me)∂2∂qi2+12∑je2ri,j−∑aZae2ri,a]Ψ+∑[−(ℏ22ma)∂2∂qa2+12∑bZaZbe2ra,b]Ψ

=iℏ∂Ψ∂t=iℏ∂Ψ∂t

If the Hamiltonian operator contains the time variable explicitly, one must solve the time-dependent Schrödinger equation. If the Hamiltonian operator does not contain the time variable explicitly, one can solve the time-independent Schrodinger equation.


Schrodinger's equation cannot be derived from anything. It is as fundamental and axiomatic in Quantum Mechanics as Newton's Laws is in classical mechanics.On scrutinizing the definition, you will find that the relation H=T+V being used is nothing but the energy conservation principle. So, from a quantum viewpoint, Schrodinger's equation is based on the energy conservation principle. Just as people have no proof for energy conservation except experiments that always appear to satisfy it, Schrodinger's equation has no pen-and-paper proof. The only proof of its validity is experiments that have never violated the equation to date.