Schrodinger Wave 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 depth.

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.

The Schrodinger Equation comes up as a mathematical expression. It describes the transformation of the physical quantity overtime, where the quantum effects are like a wave-particle duality. The equation has two forms, the time-independent Schrodinger equation and the time-dependent Schrodinger equation. The time independent equation can be easily derived from the time dependent equation, indicating that these are not two distinct, independent equations. Following that, we'll explain how the second equation follows from the first and how the first time-dependent equation can be "derived."

The Time-Dependent Schrodinger Equation

In the discussion of the particle in an infinite potential well, the wave function of a particle with fixed energy E was found to most naturally be expressed as a linear combination of wave functions with the following form:

${{\psi(x,t)} = A{e^{i(kx-\omega t)}}}$ ……(1)

Where, $A$ is the normalization constant, ${\omega}$ is the Angular frequency, $k$ is the wavenumber or the propagation constant associated with the wave and is given by ${{2{\pi}}/{\lambda}}$, $t$ is the time taken by the particle.

In order to create a standing wave and satisfy the boundary criteria, this is a representation of a wave moving in the positive x direction and a similar wave moving in the opposite direction. This indicates that we select the wave function described above as the proper wave function for a free particle because it intuitively relates to our traditional idea of a particle bouncing back and forth between the boundaries of the potential well. The free particle has momentum $p$ = ${\hbar k}$ and energy $E$ = ${\hbar \omega }$.

(Here, ${\hbar}$ is the Correction to the Planck’s constant and is equal to ${h/{2\pi}}$ and ${\omega}$ is the Angular frequency.)

With this in mind, we can then note that

${\dfrac{\partial^2 \psi}{\partial x^2}}$ = ${-{k^2}\psi}$ ……(2)

(The above equation can be obtained by double differentiating the 1st equation w.r.t. x)

Now, multiplying both the sides with (${-\frac{h^2}{2m}}$)

We will get, ${{{-\dfrac{h^2}{2m}}}{{\dfrac{\partial^2 \psi}{\partial x^2}}}}$ = ${{\dfrac{h^2}{2m}}{{k^2}\psi}}$

As we know E = ${\dfrac{p^2}{2m}}$ = ${\dfrac{{{\hbar}^2}{k^2}}{2m}}$

So, ${{{-\dfrac{h^2}{2m}}}{{\dfrac{\partial^2 \psi}{\partial x^2}}}}$ = ${\dfrac{p^2}{2m}}\psi$ …(3)

Similarly,${\dfrac{\partial \psi}{\partial t}}$ = ${-i\omega\psi}$ ….(4)

(The above equation can be obtained by one time differentiating the 1st equation w.r.t. t)

Now, multiplying both the sides with ${i\hbar}$

${{i\hbar}{\dfrac{\partial \psi}{\partial t}}}$ = ${{i\hbar}{(-i\omega\psi)}}$

As we know, E = ${\hbar\omega}$

So, ${i\hbar}{\dfrac{\partial \psi}{\partial t}}$ = ${\hbar\omega\psi}$ = ${E\psi}$ ….(5)

We now generalize this to the situation in which there is both a kinetic energy and a potential energy present, then, Total energy, ${E = K.E + P.E = {\dfrac{p^2}{2m}} + V (x)}$

Multiplying both sides with ${\psi}$,

${{E\psi} = {\dfrac{p^2}{2m}}{\psi} + {V (x)}{\psi}}$ ….. (6)

Here, ${\psi}$ is the wave function of a particle moving in the presence of a potential ${V(x)}$.

Now taking results from equation (3) and (5) we can write the equation (6) as,

${{{{-\dfrac{h^2}{2m}}}{{\dfrac{\partial^2 \psi}{\partial x^2}}}} + {V (x)}{\psi} = {i\hbar}{\dfrac{\partial \psi}{\partial t}}}$ …..(7)

Which is the famous time dependent Schrodinger wave equation. It is setting up and solving this equation, then analyzing the physical contents of its solutions that form the basis of that branch of quantum mechanics known as wave mechanics.

The Time Independent Schrodinger Equation

Though we did not arrive at that conclusion by solving the Schrodinger equation, we have seen what the wave function looks like for a free particle of energy $E$, one or the other of the harmonic wave functions, and we have seen what it looks like for the particle in an infinitely deep potential well. But in both cases, the time dependence entered into the wave function via a complex exponential factor $\left[{e^{{−iEt}/{\hbar}}}\right]$. This implies that in order to "extract" this time dependence, we should try to solve the Schrodinger wave equation using a solution of the form:

${{\psi(x,t)} = {\psi(x)}{e^{{−iEt}/{\hbar}}}}$

i.e. where the whole wave function's space and time dependencies are included in separate factors (A solution of this kind may be demonstrated to arise via the approach of "the separation of variables," a well-known mathematical technique used to solve equations of the Schrodinger equation's form). The idea now is to see if this guess enables us to derive an equation for ${\psi(x)}$, the spatial part of the wave function.

If we substitute this trial solution into the Schrodinger wave equation [equation (7)], and make use of the meaning of partial derivatives, we get:

${{{{-\dfrac{h^2}{2m}}}{{\dfrac{\partial^2 \psi (x)}{\partial x^2}}}}{e^{{−iEt}/{\hbar}}} + {V (x)}{\psi (x)}{e^{{−iEt}/{\hbar}}} = {i\hbar}{\dfrac{\partial }{\partial t}} [{\psi(x)}{e^{{−iEt}/{\hbar}}}]}$

${{{{-\dfrac{h^2}{2m}}}{{\dfrac{\partial^2 \psi (x)}{\partial x^2}}}}{e^{{−iEt}/{\hbar}}} + {V (x)}{\psi (x)}{e^{{−iEt}/{\hbar}}} = {i{\hbar}}.{{\dfrac{-iE}{\hbar}}{e^{{−iEt}/{\hbar}}}}{\psi (x)}}$

${{{{-\dfrac{h^2}{2m}}}{{\dfrac{\partial^2 \psi (x)}{\partial x^2}}}}{e^{{−iEt}/{\hbar}}} + {V (x)}{\psi (x)}{e^{{−iEt}/{\hbar}}} = E {\psi (x)}{e^{{−iEt}/{\hbar}}}}$

We now see that the factor $\left[{e^{{−iEt}/{\hbar}}}\right]$ cancels from both sides of the equation, giving us

${{{{-\dfrac{h^2}{2m}}}{{\dfrac{\partial^2 \psi (x)}{\partial x^2}}}} + {V (x)}{\psi (x)} = E{\psi (x)}}$

After rearranging the terms we will get the final equation as:

${{{{-\dfrac{h^2}{2m}}}{{\dfrac{\partial^2 \psi (x)}{\partial x^2}}}} + (E - V (x)){\psi (x)} = 0 }$

which is the time independent Schrodinger equation.

We should point out that the parameter $E$, which we have defined as the particle's energy, is a free parameter in this equation. In other words, the range of $E's$ possible values has never been limited. Therefore, all we need to do to find the wave function for a particle moving in the presence of a potential $V(x)$ and have any specific value of $E$ is to put this value of $E$ into the equation with the proper $V(x)$ and solve for the wave function. As a result, we discover—perhaps not surprisingly—that for various choices of $E$, we obtain various solutions for ${\psi (x)}$.