Question

6. In quantum mechanics, a non-relativistic particle of mass m, momentum p, and energy E, moving in the x-direction, is represented as a wave function of the form ?(x, t) = ?0 cos(kx ? ?t). Here, p = ? k, E = ? ?, and ? is Planck's constant divided by 2?. The dispersion relation for particle waves is ? = ?/(2m) k^2. (a) Express the phase velocity and group velocity of the particle in terms of its momentum and mass. Which of these expressions matches the classical relationship between velocity, momentum, and mass? (b) A particle which is known to be localized in a region of initial spatial extent ?x is represented by a wave packet that is only non-zero in this region. Demonstrate that such a wave packet disperses in such a manner that its spatial extend doubles on a time-scale which is of order (?x)^2 m/?.

          6. In quantum mechanics, a non-relativistic particle of mass m, momentum p, and energy E, moving in the x-direction, is represented as a wave function of the form
?(x, t) = ?0 cos(kx ? ?t).
Here, p = ? k, E = ? ?, and ? is Planck's constant divided by 2?. The dispersion relation for particle waves is
? = ?/(2m) k^2.
(a) Express the phase velocity and group velocity of the particle in terms of its momentum and mass. Which of these expressions matches the classical relationship between velocity, momentum, and mass?
(b) A particle which is known to be localized in a region of initial spatial extent ?x is represented by a wave packet that is only non-zero in this region. Demonstrate that such a wave packet disperses in such a manner that its spatial extend doubles on a time-scale which is of order (?x)^2 m/?.

6. In quantum mechanics, a non-relativistic particle of mass m, momentum p, and energy E, moving in the x-direction, is represented as a wave function of the form
?(x, t) = ?0 cos(kx ? ?t).
Here, p = ? k, E = ? ?, and ? is Planck's constant divided by 2?. The dispersion relation for particle waves is
? = ?/(2m) k^2.
(a) Express the phase velocity and group velocity of the particle in terms of its momentum and mass. Which of these expressions matches the classical relationship between velocity, momentum, and mass?
(b) A particle which is known to be localized in a region of initial spatial extent ?x is represented by a wave packet that is only non-zero in this region. Demonstrate that such a wave packet disperses in such a manner that its spatial extend doubles on a time-scale which is of order (?x)^2 m/?.

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