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 the Time Independent Schrodinger Equation and why is it important in quantum mechanics?
The Time Independent Schrodinger Equation (TISE) describes the allowed energy states of a quantum system where the potential does not change with time. It is a fundamental equation in quantum mechanics and helps predict properties like energy levels and wavefunctions of particles, forming the basis for understanding atomic and molecular structures as per the CBSE 2025–26 syllabus.
2. How is the Time Independent Schrodinger Equation derived from the Time Dependent Schrodinger Equation?
The TISE is derived by applying the separation of variables technique to the Time Dependent Schrodinger Equation when the potential energy is independent of time. By assuming the wave function can be written as the product of spatial and time-dependent parts, and considering stationary states, the time part factors out, leaving an equation only in terms of position.
3. What is the physical interpretation of the wave function Ψ(x) in the context of the Time Independent Schrodinger Equation?
The wave function Ψ(x) represents the probability amplitude for finding a particle at position x. The square of its absolute value, |Ψ(x)|2, gives the probability density. This means it predicts where a quantum particle is most likely to be found, which is central to understanding quantum mechanical systems in the CBSE syllabus.
4. What are the main assumptions and limitations of the Time Independent Schrodinger Equation?
The TISE assumes the potential energy is not a function of time (i.e., the system is stationary). This means it cannot describe systems with time-varying potentials or transitions between energy levels caused by external disturbances. It's most applicable to bound-state problems like electrons in an atom.
5. Can you explain the significance of the normalization constant in a quantum mechanical wave function?
The normalization constant ensures that the total probability of finding the particle anywhere in space is exactly one. Mathematically, this is achieved by integrating |Ψ(x)|2 over all space and setting it equal to one. This requirement is crucial for all quantum mechanical solutions and is a common step in CBSE board questions.
6. Why is the Time Independent Schrodinger Equation considered analogous to Newton's laws in classical mechanics?
Just as Newton's laws describe the motion of classical systems, the TISE governs the allowed energy states and spatial behavior of quantum systems. This analogy helps students understand the predictive role of the TISE, though the results are probabilistic rather than deterministic.
7. How do boundary conditions affect the solutions to the Time Independent Schrodinger Equation?
Boundary conditions specify the behavior of the wave function at the edges of the considered region (such as an infinite potential well where Ψ=0 at the boundaries). Applying suitable conditions leads to quantized energy levels and unique solutions, which is a key lesson point in the CBSE Physics curriculum.
8. What happens if the potential energy is a function of time? Can the Time Independent Schrodinger Equation still be used?
If the potential energy varies with time, the TISE is no longer valid. In such cases, the Time Dependent Schrodinger Equation must be used, as it fully incorporates the effects of time-dependent interactions and transitions between quantum states.
9. How does the concept of matter waves relate to De Broglie’s hypothesis in the context of Schrodinger’s equation?
According to De Broglie’s hypothesis, every particle in motion is associated with a wave (matter wave). Schrodinger’s equations provide a mathematical framework to describe these matter waves using the wave function, linking wave and particle behavior seamlessly in quantum mechanics, as per the CBSE pattern.
10. What are some common misconceptions students have about the Time Independent Schrodinger Equation?
Common misconceptions include thinking the equation applies to all quantum systems (it does not apply to systems with time-dependent potentials), or misunderstanding the physical meaning of the wave function as a real, physical wave instead of a probability amplitude. Clarifying these points is essential for exam success in class XII Physics.
