174x Filetype PPT File size 0.20 MB Source: atlas.physics.arizona.edu
Schrodinger Equation We take Schrodinger’s equation as one of the postulates of quantum mechanics Schrodinger himself just “figured it out” Thus there is no formal proof We rely on comparison of its predictions with experiment to validate it But we’ll briefly try to motivate it 2 Schrodinger Equation We’d like the quantum wave equation To be consistent with de Broglie-Einstein relations To be consistent with E = T+V = 2 p /2m+V To be linear in Ψ(x,t) This means if Ψ and Ψ are solutions, then 1 2 c Ψ + c Ψ is a solution 1 1 2 2 To have traveling wave solutions for a free particle (the case where V(x,t)=0) 3 Schrodinger Equation The first two assumptions can be combined into2 Ep V 2m 2k2 V 2m The third assumption means that the wave equation can only contain terms like Ψ or its derivatives (no constants or higher order powers) 4 Schrodinger Equation Recall some of our solutions to the classical wave equation sin(kx t) or eikxt Note that 2 gives a factor of k2 x2 gives a factor of t Thus we might guess a wave equation that looks like 2 V t x2 5 Schrodinger Equation We could evaluate the constants α and β using the exponential free particle solution and find (x,t) 2 2(x,t) i t 2m x2 V(x)(x,t) But we normally take Schrodinger’s equation as one of the postulates of quantum mechanics 6
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