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Lect. 1 Schrodinger equation in time dependent form
- 1. Schrodinger Equation Intime dependent Form Dr. L. S. Ravangave Associate Professor and Head Department of Physics Shri Sant Gadge Maharaj Mahavidyalaya Loha. District Nanded Maharashtra.
- 2. The wave function𝟁 is complex quantity can be represented in positive X direction as 𝟁 = A𝒆 − 𝒊 ℏ (𝑬𝒕−𝑷𝒙) (1) The equation represents the plane wave function for freely moving particle of total energy E and momentum P. To study the motion of restricted particle interacted by some kind of interaction equation (1) is insufficient.
- 3. So we derivethe fundamental second order differential equation called Schrodinger's equation. Let us start from equation. Differentiate equation (1) with respect to x twice and once with respect to t. 𝝏𝟐𝟁 𝝏𝒙𝟐 = −𝑝2 ℏ2 A𝒆 − 𝒊 ℏ (𝑬𝒕−𝑷𝒙) 𝝏𝟐𝟁 𝝏𝒙𝟐 = −𝑝2 ℏ2 𝟁, Where 𝟁= A𝒆 − 𝒊 ℏ (𝑬𝒕−𝑷𝒙) Or 𝑷 𝟐 𝟁= - 𝟏 ℏ 𝟐 𝝏𝟐𝟁 𝝏𝒙𝟐 (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 (3) The total energy (E) of the particle can be written as T.E. = E = K.E. + P.E. E= 1 2 𝑚𝑣2 + V (x) = 1 2 𝑚2 𝑣2 𝑚 +V(x) = 𝑝2 2𝑚 +V(x) Where 𝑝2 = 𝑚2 𝑣2
- 5. Multiplying by 𝟁thought E 𝟁= 𝑝2 2𝑚 𝟁 +V(x) 𝟁 (4) Substitute in equation (4) for E𝟁 and 𝑷 𝟐 𝟁 from (2) and (3) -( ℏ 𝑖 ) 𝜕𝟁 𝜕𝑡 = −ℏ 𝟐 𝟐𝒎 𝝏𝟐𝟁 𝝏𝒙𝟐 + V 𝟁 iℏ 𝜕𝟁 𝜕𝑡 = −ℏ 𝟐 𝟐𝒎 𝝏𝟐𝟁 𝝏𝒙𝟐 + V 𝟁 (5) Equation (13) is the one dimensional time dependent Schrodinger's equation. The same can be written in three dimension as iℏ 𝜕𝟁 𝜕𝑡 = −ℏ 𝟐 𝟐𝒎 𝝏𝟐𝟁 𝝏𝒙𝟐 + 𝝏𝟐𝟁 𝝏𝒚𝟐 + 𝝏𝟐𝟁 𝝏𝒛𝟐 + V 𝟁
- 6. OR iℏ 𝜕𝟁 𝜕𝑡 = −ℏ 𝟐 𝟐𝒎 𝝏𝟐 𝝏𝒙𝟐 + 𝝏𝟐 𝝏𝒚𝟐 + 𝝏𝟐 𝝏𝒛𝟐 𝟁+ V 𝟁 iℏ 𝜕𝟁 𝜕𝑡 = −ℏ 𝟐 𝟐𝒎 𝛻 𝟐 𝟁 + V 𝟁 (6) Where 𝛻2 = 𝝏𝟐 𝝏𝒙𝟐 + 𝝏𝟐 𝝏𝒚𝟐 + 𝝏𝟐 𝝏𝒛𝟐 is called Laplacian Operator Equation (6) is Time dependent form of Schrodinger's equation in three dimension Reference: Perspectives of Modern Physics By Aurthor Baiser