Time Dependent Schrodinger Equation
When many physicists were developing theories regarding quantum mechanics and the quantum phenomenon, it became a tedious part when they have to elaborate the conceptual parts in terms of mathematics. Schrodinger plays an important role in developing quantum mechanics, his theories and equations hold good and govern almost everything and every function of quantum mechanics. Schrodinger developed mathematical equations that gave a description of the wave function and helped in order to calculate the energy associated with the particle.
In 1926, Erwin Schrodinger brought up very important results of quantum mechanics, known as the Time Dependent Schrodinger Equation. The Schrodinger equations are the linear partial differential equations representing the motion of a wave function. Schrodinger developed two of them, Schrodinger Time Independent Equation and Schrodinger Time Dependent Equation and for simplification, these two equations are abbreviated as TISE and TDSE. Schrodinger’s equations in Quantum Physics are just the counterpart of Newton’s second law of motion in classical mechanics.
Time Independent Schrodinger Equation
Before going ahead with the Time Independent Schrodinger Equation derivation, we must have a clear idea of how the equation of motion is derived with respect to time. Thus, the Time Independent Schrodinger Equation derivation is completely dependent on the Time Dependent Schrodinger Equation derivation. The Schrodinger Time Dependent Equation or the Time Dependent Schrodinger Wave Equation is considered to be analogous to the classical second law of motion.
Schrodinger Time Dependent Equation
Let us begin with the Time Dependent Schrodinger Wave Equation derivation. Consider an object or the particle of mass m moving in positive x-direction with velocity v. According to De Broglie hypothesis, every particle which is under motion, there will be a wave associated with the particle which is known as the matter waves whose wavelength is given by:
\[\lambda =\frac{h}{mv}\]
\[\lambda =\frac{h}{p_{x}}\]
\[p_{x} =\frac{h}{\lambda}=\frac{h}{2\pi }\times \frac{2\pi }{\lambda }=\bar{h}k\]
\[p_{x}=\bar{h}k\]……..(1)
Where,
ħ-Correction to the Planck’s constant and is equal to\[ \frac{h}{2\pi }\]
k-Wave number associated with the wave and is given by \[ \frac{2\pi }{\lambda }\]
According to the postulates of quantum mechanics, a wave consists of tiny packets of energy known as the photon and each photon carries an energy of h.The energies of each photon is given by:
\[E=hv=\frac{h}{2\pi }\times 2\pi v=\bar{h\omega }\]…..(2)
Where,
-The angular frequency
ħ-The correction to Planck’s constant
-Angular frequency
Now, the wave function associated with the particle can be expressed by a wave function (x,t)and is given by:
\[\Psi (x,t)=Ae^{-i(\omega t-kx)}\]…..(3)
Where,
A-The normalisation constant
-Angular frequency
k-Wave number or the propagation constant associated with the wave and is given by \[ \frac{2\pi }{\lambda }\]
t-The time taken by the particle
From equation (1) and (2) we have,
\[k=\frac{p_{x}}{\bar{h}}\]….(4)
And,
\[\omega =\frac{E}{\bar{h}}\]…..(5)
Substituting the values of k and in equation (3), and on re-arranging the terms we get:
\[\Psi (x,t)=Ae^{-i(\frac{E}{h}t-\frac{p_{x}}{h}x)}\]
\[\Psi (x,t)=Ae^{\frac{-i}{h}(Et-p_{x}x)}=Ae^{\frac{i}{h}(p_{x}x-Et)}\]……(6)
Now, the total energy of the particle is given by,
E = Potential + Kinetic energy
\[E=V+\frac{p^{2}}{2m}\]
Multiplying on both sides,
\[E\Psi =(\frac{p^{2}}{2m}+V)\Psi \]…..(7)
Differentiating, equation (6) w.r.t time and hence we get,
\[\frac{d\Psi }{dt}=\frac{1}{i\bar{h}}E\Psi \]…..(8)
Similarly, differentiating equation (6) with respect to position x and hence we write:
\[\frac{d\Psi }{dx}=\frac{i}{\bar{h}}p_{x}\Psi \]……(9)
Differentiating equation (9) with respect to x:
\[\frac{d^{2}\Psi }{dx^{2}}=\frac{i}{\bar{h}}p_{x}\frac{d\Psi }{dx}\]
\[\frac{d^{2}\Psi}{dx^{2}}=\frac{i}{\bar{h}}p_{x}(\frac{i}{\bar{h}}p_{x}\Psi)=\frac{i}{\bar{h}}p_{x}^{2}\]…..(10)
Substituting equation (8) and (10) in (7) and then on simplification we get:
\[i\bar{h}\frac{d\Psi }{dt}=\frac{\bar{h^{2}}}{2m}\frac{d^{2}\Psi }{dx^{2}}+V\Psi \]…..(11)
Equation (11) is known as the Time Dependent Schrodinger Equation and it is the Time Dependent Schrodinger Equation derivation.
Time Independent Schrodinger Equation Derivation
Time Independent Schrodinger Equation is derived by considering the time t=0. Let us begin with the Time Independent Schrodinger Equation derivation, consider an object or the particle of mass m moving in positive x-direction with velocity v. According to De Broglie hypothesis, every particle which in under motion, there will be a wave associated with the particle which is known as the matter waves whose wavelength is given by:
\[\lambda =\frac{h}{mv}\]
\[\lambda =\frac{h}{p_{x}}\]
\[p_{x} =\frac{h}{\lambda}=\frac{h}{2\pi }\times \frac{2\pi }{\lambda }=\bar{h}k\]
\[p_{x}=\bar{h}k\]……..(1)
Where,
ħ-Correction to the Planck’s constant and is equal to \[ \frac{h}{2\pi }\]
k-Wave number associated with the wave and is given by \[ \frac{2\pi }{\lambda }\]
According to the postulates of quantum mechanics, a wave consists of tiny packets of energy known as the photon and each photon carries an energy of h.The energies of each photon is given by:
\[E =hv=\frac{h}{2\pi }\times 2\pi v =\bar{h}\omega \]……..(2)
Where,
-The angular frequency
ħ-The correction to the Planck’s constant
-Angular frequency
Now, the wave function associated with the particle can be expressed by a wave function (x,t)and is given by:
\[\Psi (x,0)=Ae^{i(kx-\omega t)}\]……..(3)
Where,
A-The normalisation constant
-Angular frequency
k-Wave number or the propagation constant associated with the wave and is given by \[ \frac{2\pi }{\lambda }\]
t-The time taken by the particle
Substituting the values of k and in equation (3), and on re-arranging the terms we get:
\[\Psi (x,t)=Ae^{-i(\frac{E}{h}t-\frac{p_{x}}{h}x)}\]
\[\Psi (x,t)=Ae^{\frac{-i}{h}(Et-p_{x}x)}=Ae^{\frac{i}{h}(p_{x}x-Et)}\]………(4)
We know that the Time Dependent Schrodinger Equation is given by:
\[i\bar{h}\frac{d\Psi }{dt}=\frac{\bar{-h^{2}}}{2m}\frac{d^{2}\Psi }{dx^{2}}+V\Psi \]…(5)
Differentiating, equation (4) w.r.t time and hence we get,
\[\frac{d^{2}\Psi }{dx^{2}}=\frac{1}{i\bar{h}}E\Psi \]
\[E\Psi =i\bar{h}\frac{d\Psi }{dt}\]…..(6)
Let the potential energy of the particle be dependent only on the position of the particle and independent of the time, then the wave function can be written by using the method of separation of variables and we get:
=\[\Psi (x,t)=\phi (x)u(t)\]
\[\Psi=\phi u\]………(7)
Differentiating equation (7) w.r.t x we get:
\[\frac{d\Psi }{dx}=u\frac{d\phi }{dx}\]
\[\frac{d^{2}\Psi }{dx^{2}}=u\frac{d^{2}\phi }{dx^{2}}\]….(8)
Substituting equation (7) and (8) in equation (5)
\[E\phi u=\frac{\bar{-h^{2}}}{2m}u\frac{d^{2}\phi }{dx^{2}}+V\phi u \].. (9)
Divide by u on both sides and simplifying we get:
\[E\phi=\frac{\bar{-h^{2}}}{2m}\frac{d^{2}\phi}{dx^{2}}+V\phi\]
\[\frac{\bar{-h^{2}}}{2m}\frac{d^{2}\phi }{dx^{2}}+(E-V)\phi =0 \]……(10)
Equation (10) is known as the Time Independent Schrodinger Equation in One Dimension and it is the Time Independent Schrodinger Equation derivation. The TISE in three dimensions is given by:
\[\frac{\bar{-h^{2}}}{2m}\Delta ^{2}\phi +(E-V)\phi =0 \]
Examples
Given that wavefunction of a particle is \[\Psi =A sin\frac{2\pi x}{a}\] , if the wavefunction is normalised then determine the normalisation constant.
Sol:
Given,
The wave function of the particle\[\Psi =A sin\frac{2\pi x}{a}\]
Where A is the normalisation constant and we are asked to determine its value.
We know that the normalisation constant can be evaluated by using the equation:
\[\int_{0}^{a}\Psi ^{*}\Psi =1\]
\[A^{2}\int_{a}^{0}sin^{2}\frac{2\pi x}{a}=1\]
By solving the above integration we find that,
\[A=\sqrt{\frac{2}{a}}\]
Therefore, the value of the normalisation constant is\[\sqrt{\frac{2}{a}}\]Given that wavefunction of a particle is \[\Psi =A sin\frac{3\pi x}{2a}\]
if the wavefunction is normalised then determine the normalisation constant.
Sol:
Given,
The wave function of the particle\[\Psi =A sin\frac{2\pi x}{a}\]
Where A is the normalisation constant and we are asked to determine its value.
We know that the normalisation constant can be evaluated by using the equation:
\[\int_{0}^{a}\Psi ^{*}\Psi =1\]
\[A^{2}\int_{a}^{0}sin^{2}\frac{3\pi x}{2a}=1\]
By solving the above integration we find that,
\[A=\sqrt{\frac{1}{a}}\]
Therefore, the value of the normalisation constant is \[A=\sqrt{\frac{1}{a}}\]
FAQs on Quantum Mechanics Physics/Time Independent Schrodinger Equation
1. What is Time Dependent Schrodinger Equation?
Basically, the Schrodinger equations are just the equations of motion used in Quantum Physics. They are the mathematical description for the wave motion. The Time Dependent Schrodinger Equation or the Time Dependent Schrodinger Wave Equation is considered to be analogous to the classical second law of motion.
2. What is the Time Dependent Wave Equation for a free particle?
The Time Dependent Wave Equation for a free particle is given by,
\[i\bar{h}\frac{d\Psi }{dt}=\frac{\bar{-h^{2}}}{2m}\frac{d^{2}\Psi }{dx^{2}}+V\Psi \]
Where,
m-The mass of the particle
ħ-The correction to Planck’s constant
-The wave function associated with the particle
V-The potential energy of the particle
