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Schrodinger's time independent wave equation
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- 2. SCHRÖDINGER’S WAVE EQUATION Schrödinger’sTime Dependent Equation In most situations, potential energy is a function of position as well as time. (- ℏ𝟐 𝟐𝒎 𝛁𝟐 + 𝑽)𝜳(𝒙, 𝒚, 𝒛, 𝒕) = 𝒊ℏ 𝜕𝛹 𝜕𝑡 Schrödinger’s Time Independent Equation In several situations, potential energy is a function of position alone and doesn’t depend upon time. Schrödinger’s time independent equation is also known as the steady state equation. (- ℏ𝟐 𝟐𝒎 𝛁𝟐 + 𝑽)𝜳(𝒙, 𝒚, 𝒛) = 𝑬𝜳(𝒙, 𝒚, 𝒛) Schrödinger’s equation is a wave equation in terms of the variable 𝛹 and represents the fundamental equation in quantum mechanics.
- 3. DERIVATION Consider a systemof stationary waves associated with a particle moving with velocity, ‘v’ and 𝛹 = 𝛹 (x, y, z, t) be the wave function. The differential equation of the 3-D wave motion is given as: 𝜕2 𝛹 𝜕𝑡2 = 𝑣2 𝜕2 𝛹 𝜕𝑥2 + 𝜕2 𝛹 𝜕𝑦 2 + 𝜕2 𝛹 𝜕𝑧2 = 𝑣2∇2𝛹 The solution of the above equation is of the form: 𝛹 = 𝛹0 sin 𝜔𝑡 = 𝛹0 sin 2𝜋𝜃𝑡 ⇒ 𝜕𝛹 𝜕𝑡 = 𝛹0 2𝜋𝜃 cos 2𝜋𝜃𝑡
- 4. 𝜕2𝛹 𝜕𝑡2 = −𝛹02𝜋𝜃 2 sin 2𝜋𝜃𝑡 = −4𝜋2 𝜃2 𝛹 = −4𝜋2 𝜈2 𝜆2 𝛹 On equating equations (1) and (3), we get, 𝑣2 ∇2 𝛹 = −4𝜋2 𝜈2 𝜆2 𝛹 ⇒ 𝑣2 ∇2 𝛹 + 4𝜋2 𝜈2 𝜆2 𝛹 = 0 ∇2 𝛹 + 4𝜋2 𝜆2 𝛹 = 0 ⇒ ∇2 𝛹 + 4𝜋2 𝑚2𝜈2 ℎ2 𝛹 = 0 {From 𝜆 = ℎ 𝑚𝑣 } Now, E = 1 2 𝑚𝜈2 + 𝑉 ⇒ 1 2 𝑚𝜈2 = E-V ⇒ 𝑚2 𝜈2 = 2m(E-V) Substituting for 𝑚2 𝜈2 from equation (5) into (4), we have, ∇2𝛹 + 4𝜋2 ℎ2 2𝑚(𝐸 − 𝑉)𝛹 = 0 ⇒ ∇2𝛹 + 8m𝜋2 ℎ2 (𝐸 − 𝑉)𝛹 = 0
- 5. ⇒ ∇2 𝛹 + 2m ℏ2(𝐸 − 𝑉)𝛹 = 0 ⇒ - ℏ2 2𝑚 ∇2𝛹 + (𝑉 − 𝐸)𝛹 = 0 ⇒ (- ℏ𝟐 𝟐𝒎 𝛁𝟐 + 𝑽)𝜳 = 𝑬𝜳 OR 𝑯𝜳 = 𝑬𝜳 Where 𝑯 is Hamiltonian operator. For a free particle, V = 0 so the above Schrodinger equation reduces to: - ℏ𝟐 𝟐𝒎 𝛁𝟐 𝜳 = 𝑬𝜳 ⇒ ∇2𝛹 + 2mE ℏ2 𝛹 = 0 {For a free particle}
- 6. APPLICATION OF SCHRÖDINGER’S TIME INDEPENDENTWAVE EQUATION
- 7. 1. Particle ina box/ Infinite potential well: Using Schrödinger’s Time Independent Equation, we get the following wavefunction and eigen values for this case, 𝛹n(x)= 2 𝑎 sin 𝑛𝜋𝑥 𝑎 where n = 1,2,3,… 𝐸𝑛 = 𝑛2𝜋2ℏ2 2𝑚𝑎2
- 8. 2. Quantum HarmonicOscillator: Using Schrödinger’s Time Independent Equation, we get the following wavefunction and eigen values for this case, 𝐸𝑛 = 𝑛 + 1 2 ℏ𝜔
- 9. 3. Quantum MechanicalTunnelling Effect: • Quantum mechanically, if E > 𝑉0 , then there is always some probability of reflection at x = 0 and at x = a. • Also, if E < 𝑉0 , then there is always some probability of penetration into the barrier. • In this case we apply Schrödinger’s Time Independent • Equation to understand quantum mechanical tunnelling effect. V 𝒙 = 𝑽𝟎 𝟎 < 𝐱 < 𝐚 ∞ 𝐞𝐥𝐬𝐞𝐰𝐡𝐞𝐫𝐞
- 10. MCQ 1: Whichof these is the mathematical representation of Hamiltonian operator? 1) - ℏ𝟐 𝟐𝒎 ∇ + 𝑽 2) - ℏ𝟐 𝟐𝒎 𝛁𝟐 + 𝑽 3) - ℏ𝟐 𝟒𝒎 𝛁𝟐 + 𝑽 4) - ℏ𝟐 𝟒𝒎 ∇ +𝑽
- 11. MCQ 2: Whichof the following option is true? Statement 1: Schrödinger’s time independent equation is also known as the steady state equation. Statement 2: Schrödinger’s time dependent equation is also known as the steady state equation. 1) Both the statements are true. 2) Only Statement 2 is true. 3) None of them is true. 4) Only Statement 1 is true.
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