Statement: According to Heisenberg uncertainty principle, it is impossible to measure the exact position and momentum of a particle simultaneously within the wave packet.
We know, group velocity of the wave packet is given by
Where ω is the angular frequency and k is the propagation constant or wave number
But vg is equal to the particle velocity v
Thus vg = v = ∆ω/∆k (1)
Where f is the frequency
Therefore ∆ ω = 2п ∆ f (2)
Also k=2 п/λ
Since de-Broglie wavelength λ=h/p
By putting this value in equation of k, we get
Therefore ∆k=2п∆p / λ (3)
Put equations (2) and (3) in equation (1), we get
v= 2пh∆f/2п∆p =h∆f / (4)
Let the particle covers distance ∆x in time ∆t, then particle velocity is given by
v = ∆x/∆t (5)
Compare equations (4) and (5), we get
Or ∆x.∆p=h∆f ∆t (6)
The frequency ∆f is related to ∆t by relation
∆t≥ 1/∆f (7)
Hence equations (6) becomes
A more sophisticated derivation of Heisenberg’s uncertainty principle gives
Which is the expression of the Heisenberg uncertainty principle.
As the particle is moving along x-axis. Therefore, the momentum in equation (8) of Heisenberg’s uncertainty principle should be the component of the momentum in the x-direction, thus equation Heisenberg’s uncertainty principle can be written as,
Note: There can not be any uncertainty if momentum is along y direction.
Q: Why there is uncertainty in position and momentum?
Answer: Because the particle is always in disturbed state during motion. It is not possible to calculate the position and momentum of particle simultaneously.