Wave Packets

 Wave Packets

Simple Explanation :

 

- A Wave Packet Can Be Thought Of As A "Bundle" Or "Packet" Of

  Waves That Travel Together.

- Imagine Throwing A Handful Of Stones Into A Calm Pond. Each

  Stone Creates Its Own Ripples, But When They Overlap, They

  Form A Concentrated Disturbance In A Specific Area Of The

  Pond.This Concentrated Disturbance Is Similar To A Wave

  Packet.

- It Represents A Localized And Coherent Group Of Waves That

  Propagate Together.

- A Wave Packet Can Be Defined As A Function That Represents

  The Superposition Of Multiple Waves With Different

   Frequencies, Amplitudes, And Phases.

- A Wave Packet Represents A Localized And Coherent Group Of

  Waves That Travel Together,Combining Their Properties To

  Create A Concentrated Disturbance In Space And Time.A Wave

  Packet Represents A Localized And Coherent Group Of Waves

  That Travel Together, Combining Their Properties To Create A

  Concentrated Disturbance In Space And Time.

 

(1) Concept Of Wave Packet :

 

- Accroding To De-Broglie There Is Wave Associate Withe Matter

  Partical.

- Let,

    ψ = A sin (ωt - kx) = A sin (ωt - kx)

- For Phase Velocity

   ωt - kx = Constant

   ωdt - kdx = 0

   ωdt  = kdx

   dx/dt = ω/k

   v_p = ω/k....(1)

- v_p = Wave Velocity

- Eqation (1) Called Phase Velocity

- v_p =ħω/ħk

- v_p = E/P

- v_p = mc²/mv

- v_p =c²/v

- v = Represent Partical Velocity

  v_p = Wave Velocity

 

- Acoroding To Theory Of Relativity

 

  v<c = 1 < v/c

  vp = c(c/v)

  vp > c

  so, vp > c

  v<<c

  then vp >> v

 

- Phase Velocity And Partical Velocty Are Diffrent.It Means A

  Single Wave Is Not Associate With Partical.However There Is A

  Group Of Wave Associate With Particale.

  Let As Consider Two Wave,

  ψ₁ = A sin (ω₁t - k₁x)

  ψ₂ = A sin (ω₂t - k₂x)

- From Superposition

  ψ = ψ₁ + ψ₂

  ψ = A [ sin (ω1t - k1x) +  sin (ω2t - k2x) ]

      = 2A cos [ (ω₁ - ω₂ /2)t - (k₁ -k₂/2)x] sin [(ω₁ + ω₂ /2)t  -

         (k₁ + k₂/2)x]      {sin A + sin B = 2Asin (A+B/2) cos A-

                                                                                 B/2)}......(2)

- let ω₁ ≃ ω₂

        k₁ ≃ k₂

- ω₁ + ω₂ /2 = ω + ω /2 = 2ω /2 = ω

- k₁ ≃ k₂

   k₁ + k₂/2 = k₁ +k₂ /2 = 2k /2 = k

- ω₁ - ω₂ /2 = Δω

   k₁ - k₂/2 = Δk

- all thise value put eqation no. (2)

   ψ = 2A cos ( Δωt - Δkx ) sin ( ωt - kx )

 

  2A cos ( Δωt - Δkx )= Amplitude

  sin ( ωt - kx ) = Wave

 

 

 

(2) For Group  Velocity :

 

-  Δωt - Δkx = constant

   Δωdt - Δkdx = 0

   Δωdt  = Δkdx

   dx/dt = ω/k

   v_p = Δω/Δk

   v_g = dω/dk

   v_g = ω/k

 

(i) Case 1 : For Non Relativistic Motion

 

-  v_g = dω/dk

   v_g = d(ħω)/(ħdk)

        = dE/dP

   v_g = dE/dP.....(a)

- E = p²/2m

  dE/dP = p²/2m

- v_g = P/m

       = mv/m

       = v

- vg = v ---> Group Velocity Equal To Partical Velocity

 

(ii) Case 2 : For Relativistic Motion

 

- E² = p²c² +m²c⁴

  2EdE = 2PdPc² + 0

  EdE = pc²dp

  dE/dP = pc²/E

  v_g = mvc²/mc²

  v_g = v

 

- So Group Velocity Is Equal To Particale Velocity.It Means There

  Is Group Of Wave Associate With Particale That Is Know As

  Wave Packet .

  v_g = ω/k

  ω = vgk

  dω/dk = vg +  k dvp/dk

  vg = vp + kdvp/dk

  k = 2 π /λ

  dk = -(2π/λ2) dλ

  v_g = vp + (2π/λ²) * (dvp/-2π) λ² dλ

  v_g = vp - λdvp/dλ

#WavePacket

#QuantumMechanics

#WaveFunction

#SchrödingerEquation

#QuantumPhysics

#WaveParticleDuality

#UncertaintyPrinciple

#QuantumState

#QuantumSuperposition

#QuantumInterference

#GaussianWavePacket

#WavePacketPropagation

#QuantumWavePacket

#QuantumTunneling

#WavePacketSimulation