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3.22 Griffiths - 4 Matrix Algebra in Quantum Mechanics...

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4. Matrix Algebra in Quantum Mechanics Forward to Chapter Four The theoretical basis of quantum mechanics is rooted in linear algebra. Therefore, matrices play a key role, both from a theoretical and practical viewpoint. In this chapter there is a shift in notation. I will use U to indicate potential energy, so the Schrödinger equation will be ( 29 ( 29 ( 29 ( 29 2 2 , , , 2 x t i x t U x x t t m x ψ ψ ψ = - + h h . This is because V is used as something else in Datta’s book and I want to be consistent with his notation. 4.1 Vector and Matrix Representation . State variables as vectors Let’s go back to our infinite well, but this time look at a very small one, at least in terms of the number of points used to represent the well. Suppose we use only eight points, as shown in Fig. 4.1.1. Figure 4.1.1. Infinite well with just eigth points. Implicitly, cells 0 and 9 have 0 ψ = to form the edge of the well. Figure 4.1.2 shows a function that has been initialized in the well. This, too, is a sinusoid within a Gaussian pulse, but the crudeness of our model makes it difficult to recognize. 4. Matrix Formulation 9/30/2009 1
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Figure 4.1.2. A function in an eight-point infinite well. The black bars in the lower graph are the magnitudes at the various points. This is done to emphasize that each point in the well can be thought of a basis function. The fact I am writing this in a discrete form to fit in a computer means I’m actually approximating the function ( 29 ( 29 ( 29 8 1 n n x n x x n x ψ ψ ψ δ = 2245 ⋅ ∆ = - ⋅ ∆ , (4.1.1) where x is the cell size. In this viewpoint, we are regarding the functions ( 29 x n x δ - ⋅ ∆ as basis functions used to construct to function ( 29 n x ψ ⋅∆ . The n ψ are complex coefficients showing how much of each base function is used in the construction process. Therefore, I could represent the function of Fig. 4.1.2 as a vector consisting of eight complex elements representing the values. I can do this, because I know that 4 0.0008 0.0014 i ψ = - is the value at cell number 1, 4 0.331 0.574 i ψ = - is the value at cell number 4, and so on. A waveform like Eq. (4.1.1) becomes a column vector in the matrix formalism: 1 2 3 8 ... ψ ψ ψ ψ ψ =  . (4.1.2) Notice that we have written this as a “ket.” So the question is, what is a “bra?” The bra is the transpose conjugate of the ket. Transpose means the column is rewritten as a row and conjugate means each element has been replace with its complex conjugate, * * * 1 2 8 ... ψ ψ ψ ψ = . (4.1.3) The next logical question is, what is the inner product? Obviously it’s 4. Matrix Formulation 9/30/2009 2
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1 2 * * * * * * 1 2 8 1 1 2 2 8 8 8 ... ... ... ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ = = + + + We know that this inner product sums to one, because ψ is normalized. In general, the inner product of two vectors is 1 2 * * * * * * 1 2 8 1 1 2 2 8 8 8 ... ...
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