Free Particle in Quantum Mechanics

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In quantum mechanics, the wave function of a free particle is often described using plane wave solutions. These plane waves represent particles with definite momentum and energy, but they come with certain issues:

Non-Normalizability: Plane waves extend infinitely, meaning their total probability density cannot be normalized to one. This conflicts with the probabilistic interpretation of the wave function.

Lack of Localization: A plane wave implies the particle is spread out over all space, making it impossible to determine its exact position.

To address these issues, the concept of a wave packet is introduced. A wave packet is a combination of multiple plane waves with different momenta. This superposition of waves results in a localized wave function, allowing the particle to be confined to a specific region in space.

Wave packets provide a more accurate representation of free particles in quantum mechanics. They allow for a localized position, which aligns with the probabilistic nature of quantum mechanics, and they can spread over time, reflecting the inherent uncertainty in a particle's momentum and position.

00:00 Introduction
01:19 Schrodinger's Equation - Solution
09:25 Constant Probability Density of Plane waves
12:39 Velocity of Plane waves
15:01 Non-Normalizability of Plane waves
17:24 Concept of Wave Packet

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[adityanastik]

I am PG qualified in political science and international
Please.. continue sir.... due to financial problems I didn't study science but I am much more interested in Quantum mechanics....😢😢😢

[fehmi]

Please make a video on Schrödinger equation periodic potential solution, bloch function, electron bands etc.

[SergeyPopach]

Can you, please, explain the time evolution of a free particle at which the spread of Gaussian distribution increases at t>0? Why the continuous bandwidth of a linear combination of frequencies (p=hk) would expand over time (dispersion effect)? What this physically means for a particle such as electron or light waves?

[harshyadav]

Thank you sir for this beautiful explanation

[jacobvandijk]

Free particles, being plane waves, are the "elementary particles" of quantum mechanics. From them one can build more localized particles like wave packets and delta functions. The energy (E = h . f) of a free particle is well-defined because it only has 1 frequency (or wavelength). The price we pay for that is that it is completely delocalized.

[waybeyond]

it was really a great explaination for free particle!!

[VLADTHECHAD]

sir, It would be really helpful if you could make a video lecture on Particle in a simple harmonic oscillator potential.

[abandz]

If the plane wave solution is treating the particle like a wave then shouldn’t it be normal for it to get a different velocity than the particles.

[satyampanchal-]

Hi, Sir, I was actually doing something advanced and calculating the free particle kernel via path integral. The probability density I get from that Is SAME as what I got when I calculated it for a particle under a constant force..Can you comment on why the case of a free particle and a particle under a constant force is the same as far as position space probability density is concerned?

[mathvanshi]

Please make lecture series on general theory of relativity

[ANKITKUMAR-yphq]

Sir Hindi me bhi upload kariye हम आपके द्वारा दिए गए ज्ञान के प्यासे हैं, 🙏

[MichaelKingsfordGray]

How can a "free" particle be travelling?
Relative to what? The observer?
If it is relative to ANYTHING, then it is NOT free!