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Derivation of Probability Current Density and Equation of Continuity

Derivation of Probability Current Density In Quantum Mechanics, the motion of a material particle is associated with a wave function. If the probability of finding a particle in a bounded region decreases with time, the probability of finding it outside must increase by the same amount. This is described by the Probability Current Density . 1. The Schrödinger Basis We start with the Time-Dependent Schrödinger Equation (TDSE): iℏ (∂Ψ/∂t) = -(ℏ²/2m)∇²Ψ + VΨ --- (i) Taking the Complex Conjugate of equation (i): -iℏ (∂Ψ*/∂t) = -(ℏ²/2m)∇²Ψ* + VΨ* --- (ii) 2. Mathematical Manipulation To find the rate of change, we perform the following: Multiply eq (i) by Ψ* from the left. Multiply eq (ii) by Ψ from the left. Subtract the resulting equations. Using the UV derivation method for ∇·(Ψ*∇Ψ - Ψ∇Ψ*), we simplify the expression to relate it to the Equation of Continuity . 3. The Equation of Continuity In hydrodynamics, t...
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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: ...
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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: ...
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Derivation of Probability Current Density and Equation of Continuity

Derivation of Probability Current Density In Quantum Mechanics, the motion of a material particle is associated with a wave function. If the probability of finding a particle in a bounded region decreases with time, the probability of finding it outside must increase by the same amount. This is described by the Probability Current Density . 1. The Schrödinger Basis We start with the Time-Dependent Schrödinger Equation (TDSE): iℏ (∂Ψ/∂t) = -(ℏ²/2m)∇²Ψ + VΨ --- (i) Taking the Complex Conjugate of equation (i): -iℏ (∂Ψ*/∂t) = -(ℏ²/2m)∇²Ψ* + VΨ* --- (ii) 2. Mathematical Manipulation To find the rate of change, we perform the following: Multiply eq (i) by Ψ* from the left. Multiply eq (ii) by Ψ from the left. Subtract the resulting equations. Using the UV derivation method for ∇·(Ψ*∇Ψ - Ψ∇Ψ*), we simplify the expression to relate it to the Equation of Continuity . 3. The Equation of Continuity In hydrodynamics, t...
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