Average velocity of free particle

Since we know that E=ħω and p=ħk, we can use the definitions of energy and momentum to find the average velocity. In summary, the particle average velocity can be calculated by using the wave function and the relationships between energy, momentum, and velocity. By substituting these values into the equation for average momentum, we can show that Vav = ħ(k1+k2)/2m equals ω2-ω1/k2-k1.
  • #1
mihailo
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Homework Statement


The wave function is given as Ψ(x,t) = Ae^[i(k1x-ω1t)] + Ae^[i(k2x-ω2t)].
Show that particle average velocity Vav = ħ(k1+k2)/2m equals ω2-ω1/k2-k1.
Average momentum of the particle is Pav = ħ(k2+k1)/2.

Homework Equations


p = ħk
E=ħω
K = 1/2 * mV^2

The Attempt at a Solution


if we calculate |Ψ(x,t)|^2 = 2*A(1+cos((k2-k1)x-(ω2-ω1)t)) form V=ω/k we get ω2-ω1/k2-k1.
if we use average momentum Vav = Pav/m we get Vav = ħ(k1+k2)/2m.
my problem is to get mathematically form ħ(k1+k2)/2m to ω2-ω1/k2-k1
 
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  • #2
What about using the relationship between energy and momentum?
 

Related to Average velocity of free particle

What is the definition of average velocity of a free particle?

The average velocity of a free particle is the displacement of the particle divided by the time taken for that displacement. It is a measure of the average speed and direction of the particle over a specific time interval.

How is the average velocity of a free particle calculated?

The average velocity of a free particle is calculated by taking the difference between the final position and the initial position of the particle, and dividing it by the time taken for that displacement. This can be represented by the equation: average velocity = (final position - initial position) / time taken.

What is the difference between average velocity and instantaneous velocity?

Average velocity is a measure of the average speed and direction of a particle over a specific time interval, while instantaneous velocity is the velocity of the particle at a specific moment in time. Average velocity takes into account the overall motion of the particle, while instantaneous velocity only reflects the velocity at a single point.

Can the average velocity of a free particle be negative?

Yes, the average velocity of a free particle can be negative if the particle is moving in the opposite direction of the positive direction chosen as the reference point. This indicates that the particle is moving in the negative direction and its average speed is decreasing.

What is the significance of calculating the average velocity of a free particle?

Calculating the average velocity of a free particle allows us to understand and analyze the motion of the particle over a specific time interval. It can help us determine the speed and direction of the particle, as well as any changes in its motion. This information can be useful in various scientific fields, such as physics, chemistry, and biology.

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