Operators - Schrödinger equation so is This is useful in defining a probability since we would like(3.17 Given we can thus use this freedom

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Operators Notice that in deriving the wave equation we replaced the number or by a differential acting on the wave function. The energy (or rather the Hamiltonian) was replaced by an "operator", which when multiplied with the wave function gives a combination of derivatives of the wave function and function multiplying the wave function, symbolically written as (3.16) This appearance of operators (often denoted by hats) where we were used to see numbers is one of the key features of quantum mechanics. Analysis of the wave equation One of the important aspects of the Schrödinger equation(s) is its linearity. For the time independent Schrödinger equation, which is usually called an eigenvalue problem, the only consequence we shall need here, is that if is a eigen function (a solution for ) of the
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Unformatted text preview: Schrödinger equation, so is . This is useful in defining a probability, since we would like (3.17) Given we can thus use this freedom to "normalise" the wave function! (If the integral over is finite, i.e., if is ``normalisable''.) As an example suppose that we have a Hamiltonian that has the function as eigen function. This function is not normalised since (3.18) The normalised form of this function is (3.19) We need to know a bit more about the structure of the solution of the Schrödinger equation - boundary conditions and such. Here I shall postulate the boundary conditions, without any derivation. 1. is a continuous function, and is single valued. 2. must be finite, so that (3.20) 3. 4. is a probability density. 5. is continuous except where has an infinite discontinuity....
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This note was uploaded on 11/09/2011 for the course PHY PHY2053 taught by Professor Davidjudd during the Fall '10 term at Broward College.

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Operators - Schrödinger equation so is This is useful in defining a probability since we would like(3.17 Given we can thus use this freedom

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