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Scrodinger wave equation
- 1. Time Independent Schrodinger Wave Equation Dr. Sujit K. Shah, Tribhuvan University, Nepal 7/15/2020 1
- 2. Two basic observations • The wave character of the electron These observations lead us to use a wave equation and attempt to introduce particle character through the de Broglie relation • The probability character of our measurements Dr. Sujit K. Shah, Tribhuvan University, Nepal 7/15/2020 2
- 3. The general partial differential equation of wave motion is: 𝜕2 𝜓 𝜕𝑥2 + 𝜕2 𝜓 𝜕𝑦2 + 𝜕2 𝜓 𝜕𝑧2 = 1 𝑣2 𝜕2 𝜓 𝜕𝑡2 𝛻2 = 𝜕2 𝜕𝑥2 + 𝜕2 𝜕𝑦2 + 𝜕2 𝜕𝑧2 known as Laplacian operator 𝜕2 𝜕𝑥2 + 𝜕2 𝜕𝑦2 + 𝜕2 𝜕𝑧2 𝜓 = 1 𝑣2 𝜕2 𝜓 𝜕𝑡2 𝛻2 𝜓 = 1 𝑣2 𝜕2 𝜓 𝜕𝑡2 1 Dr. Sujit K. Shah, Tribhuvan University, Nepal 7/15/2020 3
- 4. The function 𝜓 can be resolved in two functions; function of space coordinates 𝜓(𝑥,𝑦,𝑧) and function of time 𝑔(𝑡) 𝜓 = 𝜓(𝑥,𝑦,𝑧) 𝑔(𝑡) Substituting this to equation 1 𝛻2 𝜓(𝑥,𝑦,𝑧) 𝑔(𝑡) = 1 𝑣2 𝜕2 𝜓(𝑥,𝑦,𝑧) 𝑔(𝑡) 𝜕𝑡2 𝑔(𝑡) 𝛻2 𝜓(𝑥,𝑦,𝑧) = 1 𝑣2 𝜓(𝑥,𝑦,𝑧) 𝜕2 𝑔(𝑡) 𝜕𝑡2 Dr. Sujit K. Shah, Tribhuvan University, Nepal 7/15/2020 4
- 5. In order to separate time dependence, several wave functions may chosen for 𝑔(𝑡), such as 𝑒(2𝜋𝑖𝜈𝑡) or 𝑆𝑖𝑛 2𝜋𝜈𝑡 If we substitute 𝑔(𝑡) = 𝑒(2𝜋𝑖𝜈𝑡) In order to separate time dependence, several wave functions may chosen for 𝑔(𝑡), such as 𝑒(2𝜋𝑖𝜈𝑡) or 𝑆𝑖𝑛 2𝜋𝜈𝑡 If we substitute 𝑔(𝑡) = 𝑒(2𝜋𝑖𝜈𝑡) 𝑒(2𝜋𝑖𝜈𝑡) 𝛻2 𝜓(𝑥,𝑦,𝑧) = 1 𝑣2 𝜓(𝑥,𝑦,𝑧) 𝜕2 𝑒(2𝜋𝑖𝜈𝑡) 𝜕𝑡2 Dr. Sujit K. Shah, Tribhuvan University, Nepal 7/15/2020 5
- 6. 𝑒(2𝜋𝑖𝜈𝑡) 𝛻2 𝜓(𝑥,𝑦,𝑧) = 1 𝑣2 𝜓 𝑥,𝑦,𝑧 (−4𝜋2 𝜈2 )𝑒(2𝜋𝑖𝜈𝑡) 𝜕𝑒(𝟐𝝅𝒊𝝂𝑡) 𝜕𝑡 = (𝟐𝝅𝒊𝝂)𝑒(𝟐𝝅𝒊𝝂𝑡) 𝛻2 𝜓(𝑥,𝑦,𝑧) = −4𝜋2 𝜈2 𝑣2 𝜓 𝑥,𝑦,𝑧 It is seen that the variable, 𝑡, cancels, and we have succeeded in separating out the time dependence leaving a wave equation that depends only on space coordinates. 2 Dr. Sujit K. Shah, Tribhuvan University, Nepal 7/15/2020 6
- 7. Equation 2 represents the wave portion. 𝛻2 𝜓(𝑥,𝑦,𝑧) = −4𝜋2 𝜆2 𝜓 𝑥,𝑦,𝑧 𝜈 = 𝑣/𝜆 The particle character is introduced using de Broglie equation: 𝜆 = ℎ 𝑚𝑣 𝛻2 𝜓(𝑥,𝑦,𝑧) = −4𝜋2 𝑚2 𝑣2 ℎ2 𝜓 𝑥,𝑦,𝑧 Dr. Sujit K. Shah, Tribhuvan University, Nepal 7/15/2020 7
- 8. Total energy 𝐸 = 𝐾. 𝐸. + 𝑉 1 2 𝑚𝑣2 = 𝐸 − 𝑉 𝛻2 𝜓(𝑥,𝑦,𝑧) = −8𝜋2 𝑚 ℎ2 (𝐸 − 𝑉)𝜓 𝑥,𝑦,𝑧 Dr. Sujit K. Shah, Tribhuvan University, Nepal 7/15/2020 8
- 9. 𝛻2 𝜓(𝑥,𝑦,𝑧) = −8𝜋2 𝑚 ℎ2 (𝐸 − 𝑉)𝜓 𝑥,𝑦,𝑧 − ℎ2 8𝜋2 𝑚 𝛻2 𝜓(𝑥,𝑦,𝑧) = (𝐸 − 𝑉)𝜓 𝑥,𝑦,𝑧 − ℎ2 8𝜋2 𝑚 𝛻2 𝜓 𝑥,𝑦,𝑧 + 𝑉𝜓 𝑥,𝑦,𝑧 = 𝐸𝜓 𝑥,𝑦,𝑧 (− ℎ2 8𝜋2 𝑚 𝛻2 + 𝑉)𝜓 𝑥,𝑦,𝑧 = 𝐸𝜓 𝑥,𝑦,𝑧 𝐻𝜓 𝑥,𝑦,𝑧 = 𝐸𝜓 𝑥,𝑦,𝑧 Dr. Sujit K. Shah, Tribhuvan University, Nepal 7/15/2020 9
- 10. 𝐻𝜓 𝑥,𝑦,𝑧 = 𝐸𝜓 𝑥,𝑦,𝑧 𝐻 = (− ℎ2 8𝜋2 𝑚 𝛻2 + 𝑉) Hamiltonian Operator An equation, where the operator, operating on a function, produces a constant times the function, is called an eigenvalue equation. The function is called an eigenfunction, and the resulting numerical value is called the eigenvalue. Eigenvalue Eigenfunction Dr. Sujit K. Shah, Tribhuvan University, Nepal 7/15/2020 10