Particle in a 1D Box | Infinite Potential Well Problem in QM

The Infinite Potential Well problem is one of the most important and simplest problems in Quantum Mechanics. In this video, I do a complete discussion on the topic.

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The particle in a box problem considers a single particle, such as an electron, trapped in a one-dimensional region between two impenetrable walls. The walls are so high that the particle cannot escape, meaning its potential energy is zero inside the box and infinitely large outside.

Wavefunction: In quantum mechanics, the state of the particle is described by a wavefunction. This wavefunction represents the probability amplitude of finding the particle at a specific position within the box. Inside the box, the wavefunction must meet certain boundary conditions: it must be zero at the walls of the box because the particle cannot exist outside the box.

Quantization: Unlike in classical mechanics, where the particle can have any energy, the particle in a box can only have specific, discrete energy levels. These energy levels are quantized and are determined by the size of the box and the mass of the particle.

Nodes and Antinodes: The wavefunction inside the box exhibits a pattern of nodes (points where the wavefunction is zero) and antinodes (points where the wavefunction has maximum amplitude). The number of nodes increases with the energy level, indicating higher energy states correspond to more complex wave patterns.

Energy Levels: The lowest energy state, called the ground state, has the simplest wavefunction with no nodes (except at the boundaries). Higher energy states, called excited states, have increasing numbers of nodes. The spacing between energy levels increases as the particle's energy increases.

This problem illustrates fundamental quantum mechanical principles, such as quantization of energy and the wave nature of particles. It also serves as a basis for understanding more complex systems in quantum mechanics, including atoms and molecules.

00:00 Introduction
01:53 Solution of Time Independent Schrodinger's Eqn
06:43 Boundary Conditions
10:48 Discrete Energy Levels
17:32 Normalization & Wavefunction
24:23 Visualization of Eigenfunction & Probabilities
30:07 Properties of Eigenfunction Sulutions

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[ajittiwari]

Please continue this series on Quantum Mechanics ❤

[pauldirac]

Another great lecture Dibs, thanks.I think that after 30 episodes, you finally found your stride.And these lectures are becoming fun and informative and exciting.Keep it up!

[koushiksaikia]

Very well explained sir.Please make a video on Dirac orthonormality related to momentum eigen functions. Also address the point, why we can't have discrete momentum eigen values. I have been trying to understand this portion from Griffiths QM book, but not understood properly.

[SergeyPopach]

In quantum physics, the particle is not always has discrete energy levels. It has only discrete energy levels if it’s in bounded states.

[jackdeago]

Thank you for such awesome videos can you made another series like this but about general relativity

[AyanMandal-pkjm]

Hi i am a very old viewer of your channel. I started watching your channel when i was in class 9 and now i have completed my 12 th .i always like your video very much.

I have a doubt in this question that why the probability density is independent of time? As the particle has a velocity it should be moving inside the well, so the probability density should vary with time but it is not happening .can you explain why this is happening?

[fehmi]

Beautiful lecture, I hope you make a video on periodic potential, Bloch function etc.

[Ankankumar]

'Sir when your new lecture is coming' it feels like Endgame is coming after Infinity War 😅

[RiyaShrivastava-wl]

Sir pls make video on chemical bonding and molecular structure

[zhenwalker]

Which book is best for general relativity

[VikramSingh-gsud]

Sir what is utility of these 1D 2D potenials n harmonic oscliiator..plz explain where i see n use these in my daily life..plz explain some physics examples where these concepts r used..plz reply sir..i would b obliged sir

[koushikmandal]

For this case Hamiltonian and momentum operator commute with each other. Can you please tell why the eigen function of the Hamiltonian is not an eigen function of the momentum operator ?

[pauldirac]

I also heard about things like the Fourier trick or the Fourier transform As a method to find the CN coefficients.I'm really interested and curious about it.If somebody could please tell me...I'm dying here..is there anybody out there?..

[SergeyPopach]

The infinite potential well is just limiting case of more realistic, but complex finite well case. Just like Dirac delta function potential case…

[pauldirac]

Are the Cn coefficients of the linear superposition found by using a Fourier series analysis? Or something else?

[pauldirac]

I'll ask again, What determines the value of the Cn coefficients? @37:40

[Gokulkrish-kb]

Energy associated with the particle in 1d box is discreate why?
Bcz in QM we represent the particle by using wave function (Wave) so the wavelength associated with the wave with in the box is discrete so the wave number K is also discrete and the engrgy also discrete ( This is only true for the bounded systems like Infinite box, LHO and H atom )
Is this correct sir .... If not please explain

[tulangsodimha]

You didn't teach how to draw wave function graph.😢

[Folorunsho]

14:56 if our system is 1-dimensional, why are the energy levels increasing in another direction?
Please explain why that's so.

[deviljetsay]

Sir please series me itna gap mat kijie😊