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Unformatted text preview: Chem 120A Some basic concepts in wave mechanics 09/02/2010 Fall 2010 Discussion 1 All of this is done for one particle in 1D but is trivially generalizable to any number of particles in any number of dimensions. Operators An operator on a function is an object which enacts some sort of transformation on the function e.g. ˆ A f ( x ) = g ( x ) (1) or (more concretely) f ( x ) = x 2 (2) ˆ A = d dx (3) ˆ A f ( x ) = 2 x = g ( x ) (4) They are completely analogous to matrices in linear algebra which enact transformations on vectors (think rotation matrices which when applied to vectors rotate the vector by some angle θ ). Like matrices, operators can have eigenvalues, i.e. they can fulfill the relationship ˆ A f ( x ) = λ f ( x ) (5) where λ is an eigenvalue of ˆ A and f ( x ) is an eigenfunction of ˆ A . Turns out, you’ve seen an eigenvalue problem of this type before! The time-independent Schr¨odinger equation is an eigenvalue problem where ˆ H ψ ( x ) = − ¯ h 2 2 m d 2 dx 2 ψ ( x )+ V ( x ) ψ ( x ) = E ψ ( x ) (6) and ˆ H is the Hamiltonian which operates on the wavefunction and returns the total energy of the system (it is a statement of the conservation of energy, ˆ T + ˆ V )....
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