8. LECT9 - Schrodinger WAVE EQN PDF

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INTRO TO SCHRODINGER WAVE EQUATION Dynamic of microscopic systems 

Quantum mechanical view: o A particle is spread through space like a wave 

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Wavefunction (Ψ) details the behaviour of an electron-wave

Probability of finding an e- at a given pt in space = ψ2

Schrodinger wave eqn (1927) 

Double partial differential eqn

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TOTAL ENERGY = KINETIC ENERGY + POTENTIAL ENERGY (V)

o ET= ½mv2 + V =

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V= potential energy

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v= velocity = distance travelled/time – ms-1

The eqn can be represented as: ΗΨ=ΕΨ o Where H= Hamiltonian operator; E= energy; Ψ= wavefunction 

Hamiltonian operator: 

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Only used in particle-in-a-box

Simple form of the Schrodinger eqn:

o 

If there is no potential energy  free particle

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If there is potential energy stored (eg. Spring)  chemical bond

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If potential energy stored in electric attraction b/t a +vely & -vely charged particle (Coulombic potential)  H-atom

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Ψ in 3 cases: 

Ψ for freely moving particle = sin(x) [sin curve]

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Ψ for particle free to oscillate near a pt = e-x^2 [bond vibrating]

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Ψ for e- in H-atom = e-r [r=distance from the nucleus]

Physical significance of Ψ 

Born interpretation: probability of finding a particle in a small region of space of volume δV = proportional to Ψ2δV o Where Ψ=wavefunction in the region

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Ψ2  probability density o Probability = probability density * volume of region of interest o Low Ψ2 value = small chance of finding the particle...