Lecture 6 - Schrodinger's Equation Time dependent SE Time...

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Schrodinger’s Equation Time dependent SE Time independent SE Particle in a box
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Wave Equation ( 29 ( 29 sin sin 2 sin x t A x vt A A kx t T ψ π ϖ λ = - = - = - The general form of the wave function is which also describes a wave moving in the x direction. In general the amplitude may also be complex. The wave function is also not restricted to being real. Notice that the sine term has an imaginary number. Only the physically measurable quantities must be real. These include the probability, momentum and energy. Harmonic plane wave traveling in the positive direction x :
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Conditions on the Wave function In order for ψ to be a solution of the Schrödinger equation to represent a physically observable system, ψ must satisfy certain constraints: 1. Must be a single-valued function of x and t ; 2. Must be normalisable; This implies that the ψ 0 as x ; 3.ψ ( x ) must be a continuous function of x ; 4. The slope of must be continuous, specifically d ( x )/d x must be continuous (except at points where potential is infinite).
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The probability of finding a particle somewhere in space must be unity , thus the normalization condition : In 1D: - a wavefunction which obeys this condition is said to be normalized ( 29 2 , 1 x t dx - Ψ = Suppose we have a solution to the Sch. Eq. that is not normalized. The recipe for normalization: Calculate the normalization integral Re-scale the wave function as 2 ( , ) d N x t x - = Ψ ( 29 ( 29 1 ' , , r t r t N Ψ = Ψ r r This procedure works because any solution of the S.Eq. being multiplied by a constant remains a solution: the S. .Eq. is linear & homogeneous . ( 29
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Lecture 6 - Schrodinger's Equation Time dependent SE Time...

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