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Schrodinger equation-using oprators
- 1. LECT. 2. OPRATORS Dr.L. S. Ravangave Associate Professor and Head Department of Physics Shri Sant Gadge Maharaj Mahavidyalaya Loha. District Nanded Maharashtra.
- 2. OPRATORS In Wave Mechanicsthe total energy E and momentum P are dynamical quantities that changes with time and expectation values of these quantities cannot be found by treating the equations. E expt= −∞ ∞ 𝟁 ∗ 𝐸𝟁 dx P expt= −∞ ∞ 𝟁 ∗ 𝑝 𝟁 dx Therefore these quantities can be expressed in the form of their differential oprators called total energy oprator and momentum oprator.
- 3. To obtain theseoprators, let us start from free particle wave function 𝟁 = A𝒆 − 𝒊 ℏ (𝑬𝒕−𝑷𝒙) (1) Differentiate equation (1) with respect to x and once with respect to t. 𝜕𝟁 𝜕𝑥 = A𝒆 − 𝒊 ℏ (𝑬𝒕−𝑷𝒙) (− 𝑖 ℏ ) (-p) 𝜕𝟁 𝜕𝑥 = - 𝑖𝑃 ℏ 𝟁 Where 𝟁= A𝑒 − 𝑖 ℏ (𝐸𝑡−𝑃𝑥) Or P𝟁= - ℏ 𝑖 𝜕𝟁 𝜕𝑥 (2)
- 4. Differentiate equation (1)with respect to t. 𝜕𝟁 𝜕𝑡 = A𝑒 −𝑖 ℏ 𝐸𝑡 − 𝑃𝑥 (- 𝑖 ℏ ) E 𝜕𝟁 𝜕𝑡 = (- 𝑖 ℏ ) E 𝟁 Or E 𝟁 =-( ℏ 𝑖 ) 𝜕𝟁 𝜕𝑡 E 𝟁= 𝑖ℏ 𝜕𝟁 𝜕𝑡 (3) Evidently we write from equation (2) and (3) P= - ℏ 𝑖 𝜕 𝜕𝑥 (5) E = 𝑖ℏ 𝜕 𝜕𝑡 (6) The equations (5) and (6) represents the Momentum and total energy oprators respectively.
- 5. We check thevalidity of these oprators by means of Schrodinger equation. The total energy (E) of the particle can be written as T.E. = E = K.E. + P.E. or E= 𝑝2 2𝑚 +V(x) Write above equation in oprator form E= 𝑷 𝟐 𝟐𝒎 +V Substitute the values of the oprators from equations (2) and (3)
- 6. 𝑖ℏ 𝜕 𝜕𝑡 = 1 2𝑚 (− ℏ 𝑖 𝜕 𝜕𝑥 )2 + V Or𝑖ℏ 𝜕 𝜕𝑡 = - ℏ2 2𝑚 𝝏 𝟐 𝝏𝒙 𝟐 + V Multiplying by 𝟁 𝑖ℏ 𝜕𝟁 𝜕𝑡 = - ℏ2 2𝑚 𝝏 𝟐 𝟁 𝝏𝒙 𝟐 + V𝟁 This is time dependent from of Schrodinger equation which gives the validity of the total energy and momentum oprators