Derivation of Time Dependent and Time Independent Schrodinger Wave Equation (Step-by-Step)

Mastering the Schrödinger Equation: Time-Dependent & Independent Forms

In quantum mechanics, the Schrödinger Equation is the cornerstone that describes the behavior of subatomic particles. Unlike classical physics, it uses wave functions to determine the probability of a particle's state. In this guide, Admin Focus breaks down the complete derivations for both forms as studied in BSc Physics.

1. Derivation of the Time-Dependent Schrödinger Equation (TDSE)

We begin with the principle of conservation of energy, where Total Energy (E) is the sum of Kinetic Energy (K.E.) and Potential Energy (P.E.):

E = (p² / 2m) + V

To transition to a wave representation, we apply this to the wave function Ψ:

EΨ = (p² / 2m)Ψ + VΨ --- (Eq. 1)

Operator Substitution

In quantum mechanics, we replace physical quantities with operators. Based on the fundamental postulates:

• Energy Operator (E): -(ℏ/i) (∂/∂t)
• Momentum Operator (p): -(ℏ/i) (∂/∂x)

Squaring the momentum operator gives us p² = -ℏ² (∂² / ∂x²). Substituting these into Eq. 1, we arrive at the final Time-Dependent form:

-(ℏ/i) (∂Ψ / ∂t) = - (ℏ² / 2m) (∂²Ψ / ∂x²) + VΨ

---

2. Derivation of the Time-Independent Schrödinger Equation (TISE)

The Time-Independent form is used when the potential V is constant over time. We start by using the Method of Separation of Variables for the wave function:

Ψ = A e ^{i/ℏ (Et - px)}

Mathematical Steps

We let the spatial part be ψ₀ = A e ^ipx/ℏ . This allows us to express the total wave function as:

Ψ = ψ₀ e ^-iEt/ℏ

By differentiating this equation with respect to t and x (twice), we find the respective derivatives for the TDSE. After substituting these values back into the Time-Dependent equation, the time-dependent exponential terms cancel out, leaving us with the spatial equation:

E ψ₀ = - (ℏ² / 2m) (∂²ψ₀ / ∂x²) + V ψ₀

(∂²ψ₀ / ∂x²) + (2m / ℏ²) (E - V) ψ₀ = 0

Conclusion & Significance

These equations allow us to solve for allowed energy levels in various systems, such as the Quantum Harmonic Oscillator or the Hydrogen Atom. Understanding the transition between the time-dependent and independent forms is essential for any BSc Physics student.

— Admin Focus