Group & Phase Velocities of Wave packet in Quantum Mechanics

Group velocity is the velocity of the quantum particle while Phase velocity is the velocity of the wavefunction. They are not equal.

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In quantum mechanics, group velocity refers to the speed at which the peak or envelope of a wavepacket propagates through space. Phase velocity, on the other hand, refers to the speed at which the phase of the wave oscillation moves. In some cases, phase velocity can exceed the speed of light, but it doesn't carry actual energy or information. Group velocity is associated with the movement of the wave packet itself, while phase velocity is related to the oscillatory behavior of the wave within the packet. In quantum mechanics, the evolution of this wave packet is an approximation of the trajectory of the classical particle.

00:00 Introduction to Wave packet
06:15 Dispersive & Non-dispersive Medium
09:45 Group & Phase velocities
26:20 Group Velocity of Particle
37:17 Phase Velocity of Particle

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[tooclose]

00:03 Wave packet describes particle motion in quantum mechanics
02:41 Wave packet represents particle's motion and its evolution through space.
07:58 Different frequencies travel at the same or different speeds based on the medium's dispersion.
10:18 Wave packet can be simplified using Taylor series expansion
15:44 The wave packet comprises a plane wave solution and a modulating function.
18:10 Comparing the speeds of plane wave and modulating function.
22:55 Group velocity represents the velocity of the entire wave packet.
25:04 Group velocity and phase velocity can be equal, unequal, or opposite.
29:48 Group velocity is related to particle energy and momentum.
32:08 In quantum mechanics, the group velocity represents the particle's velocity.
36:54 Group velocity is the same as the particle's velocity, but phase velocity may not have a physical significance.
38:53 Phase velocity is always greater than the speed of light
43:11 Wave packet idea discussed, more details next class

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[ItsDeveshA]

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[YossiSirote]

First of all, I love this video!! I am a second year grad student in physics at Tel Aviv University, and I have never heard this explained that well.
Second, why is the modulating function only a function of x? Shouldn’t it be a function of x & t?

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[aafeer]

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[SergeyPopach]

The thing is that so called "wave packet" is a summation of plane waves of varying frequencies and amplitudes, which introduces the analytical tool of integration over continuous range of k (momentum). Although it is now confined spatially (as we make its momentum more uncertain) and wavefunction becomes normalized;... it is not quantized yet. The fact that integration is being used here, we are still dealing with continuous variable of k. This means that we have a vector space with orthonormal base of infinite size of eigenstates. The result of a precise measurement on a single quantum system is always one of the eigenvalues of the measured observable (whaterever operator we apply to quantum system). If we still have infinite amount of eigenstates (eigenvectors).

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34:16 formulas to remember, for relativistic also

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