CHM 312 Week 3 The SE and Particle in a Box(3)

CHM 312 Week 3 The SE and Particle in a Box(3) - CHM 312...

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CHM 312 Quantum Mechanics Week 3: The Schrödinger Equation
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The Schrödinger Equation The Schrödinger equation (S.E.) is the fundamental equation to QM. Solutions to S.E. are called wave functions (), and provide a complete description of the system. Like Newton’s laws ( f = ma ), the S.E. is not derivable and stands as a postulate of QM. We can show it is plausible via a derivation like exercise. From de Brogile’s idea that matter acts as waves, it was postulated that there should be a wave equation that governs matter. The race was on to explain this and Schrödinger and Heisenberg came up with varying methods that resulted in the same answer in the same year.
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Towards the SE Physicists had the classical wave equation to start with: Remember, the solution to the equation above had some spatial function ( X ( x ) ) multiplied by a time function ( T ( t ) ): We will reassign X ( x ) as and call it the spatial amplitude of the displacement, . With the preceding in mind can then be written as: Remember that was related to the angular velocity of the string. i.e ., Substituting in for in the classical wave equation yields:
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Towards the SE Taking of wrt x does nothing to so it acts as a constant. Same for and . Dividing through by : Now time independent! Equation above becomes: Eigenvalue problem again
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Towards the SE From the summary of classical operators we know: 1) and 2) Substituting and simplifying: Coming back to de Broglie : Therefore: Remember that total energy ( E ) is the sum of K.E. and P.E.: Therefore:
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Time Independent S.E. Therefore: Substituting for : Since : Rearranging we get the time independent S.E.!!!
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Operat ors Classical mechanical quantities (position (), momentum (), K.E. (), P.E.(), total E (), etc…) are represented by linear operators in QM. An operator is a function (or operation) that acts on another value and/or function. e.g ., in the expression, , is the operator acting on the function, e.g ., , where is the operator acting on the function . e.g ., , where is the operator acting on the function,. e.g ., 3( x 2 ), where 3x is the operator acting on the function, x 2 . Operators can be significantly more complicated but it’s the same idea. The complex nature of some operations is why the shorthand notation of operators was invented. Operators are typically represented by a “hat” ().
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More Operator Examples 1)
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Linear Operators Said before that the operators must be linear , that means the operator must satisfy the following: Is a linear operator? Is a linear operator? Is a linear operator? and can be complex
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Eigen this Eigen that… Operator Eigenfunction Eigenvalue The SE turns out to be an eigenvalue problem and is of the form: In an eigenvalue problem, the operation yields a constant times the original eigenfunction i.e ., the eigenfunction remains the same after the operation.
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The Hamiltonian and the Eigenvalue Problem In the expression , given and a set of boundary conditions, it will be our job to
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