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The Schrodinger equation and wave-function, which are fundamental concepts in quantum mechanics. It discusses the normalization of wave-function, superposition principle, and probability interpretation. The document also provides specific examples of wave-functions for a photon, harmonic motion, and free particle. The Schrodinger equation is derived and explained in detail. suitable for students studying quantum mechanics or related fields.
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3. Normalization dx = 1 Ψ= Ψ 1 + Ψ 4. Superposition principle 2 **5 Continuous and single valued
2 dx = 1 Conservation of Probability (normalization) Schrodinger Equation Wave-function^ (complex) Probability interpretation: P ( x ) dx =|Ψ ( x , t )| 2 dx What is Schrodinger equation? Atom Harmonic motion Free particle Specific Examples
2 2 m
2 2 m
2 ∂ x 2
∂ Ψ ( x , t ) ∂ t
2 2 m
2 Ψ ( x , t ) ∂ x 2
∂ Ψ ( x , t ) ∂ t = H Ψ ( x , t ) Schrodinger Equation
For a photon, wavefunction is of the wave form Ψ ( x ,t ) = A e i ( kx − ω t ) k =
h p Ψ ( x ,t ) = A e i ( p ℏ x − ω t ) Therefore, k =
h p = p
[ we can also use the form (^) Ψ ( x ,t ) = A sin ( kx − ω t )]
Schrodinger Equation
∂ Ψ ( x , t ) ∂ t
2 2 m
2 Ψ ( x , t ) ∂ x 2
∂ Ψ ( x , t ) ∂ t = H Ψ ( x , t )
2 2 m
2 ∂ x 2
Time independent Schrodinger Equation i ℏ ∂ Ψ ( x , t ) ∂ t = − ℏ 2 2 m ∂ 2 Ψ ( x , t ) ∂ x 2
− ℏ 2 2 m ∂ 2 ψ( x ) ∂ x 2
= E
