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| classes:2009:fall:phys4101.001:lec_notes_0914 [2009/09/20 09:49] – yk | classes:2009:fall:phys4101.001:lec_notes_0914 [2009/09/20 10:23] (current) – yk |
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| //This matrix equation can be interpreted in the following way. When a matrix operate on a vector, it will result in a vector. This resulting vector in general is different from the original vector in both direction and the length.// | //This matrix equation can be interpreted in the following way. When a matrix operate on a vector, it will result in a vector. This resulting vector in general is different from the original vector in both direction and the length.// |
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| //For example, we can think of// a simple transformation //created by a familiar// 2-Dimensional rotation matrix: <math>\[\begin{array}{ccc} | //For example, we can think of// a simple transformation //created by a familiar// 2-Dimensional rotation matrix: <math>R = \[\begin{array}{ccc} |
| cos(\theta) & sin(\theta) \\ | cos(\theta) & sin(\theta) \\ |
| -sin(\theta) & cos(\theta) \end{array} \]</math> | -sin(\theta) & cos(\theta) \end{array} \]</math> |
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| //With this matrix, all vectors are rotated by an angle θ and therefore change their directions. Looking at the eigenvector equation above, it suggests that the two vectors, the original and the one after the transformation by the matrix, M, are in the same direction. //i.e.// for the matrix M, vector// <math>\psi</math> //is a special vector which does not change its direction after being operated on by the matrix, M.// <del>operator that acts on the wave function and an eigenvector, which equal an eigenvalue multiplied by the same eigenvector</del>: | //With this matrix, all vectors are rotated by an angle θ and therefore change their directions. |
| | <del>For the Eigenvalue equation, the matrix //M// does //not// change the direction of the eigenvector <math>\psi</math>, only its magnitude; it is equivalent to multiplication by a //constant//.</del> |
| | Looking at the eigenvector equation above, it suggests that the two vectors, the original and the one after the transformation by the matrix, M, are in the same direction. //i.e.// for the matrix M, vector// <math>\psi</math> //is a special vector which does not change its direction after being operated on by the matrix, M.// <del>operator that acts on the wave function and an eigenvector, which equal an eigenvalue multiplied by the same eigenvector</del>: |
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| //With this interpretation, the rotation matrix shown above cannot have eigenvectors. This is represented by the fact that this rotation matrix does NOT have real eigenvalues. i.e. there is no vectors whose directions remain the same after the rotation. However, if we expand ourselves and accept complex eigenvalues and vectors with complex components, even this matrix does have two eigen values and corresponding eigen vectors. They are: //<math> </math> | //With this interpretation, the rotation matrix, //R//, shown above cannot have eigenvectors. This is represented by the fact that this rotation matrix does NOT have real eigenvalues. i.e. there is no vectors whose directions remain the same after the rotation. However, if we expand ourselves and accept complex eigenvalues and vectors with complex components, even this matrix does have two eigen values and corresponding eigen vectors. They are: //<math>\cos(\theta)\pm i\sin(\theta)={\rm e}^{\pm i\theta},</math> and <math>\[\begin{array}{c} 1 \\ \pm i \end{array} \] </math> |
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| <math>H\psi = E\psi</math> | <math>H\psi = E\psi</math> |
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| We know from Linear Algebra that an //n// dimensional matrix //M// and the Eigenvalue/vector equation can be solved for <math>(n-1)</math> variables and <math>\lambda</math>. Multiplication by the matrix M represents a linear transformation of <math>\psi</math>, and the eigenvalue equation represents a transformation that maps all values of <math>\psi</math> to zero. | //It may seem strange to be able to figure out what the unknown wave function AND unknown energy value from a single equation. It almost looks like two unknowns can be figured out from one equation. For the eigenvector equation, the same thing is happening. Unknown eigenvector has //n// unknowns, in addition to an additional unknown eigenvalue. So altogether there are //n+1// unknowns whereas the matrix equation represents //n// equations. How is this possible? Doesn't it go against the basic of algebra? It turns out that we are not able to determine everything about the eigenvector. We can determine only its direction, but not the length. The length must be determined by other means. For our Schrödinger equation, the same thing happens - the normalization of the wave function is NOT determinable. But our normalization requirement arising from the requirement that the total probability has to be 1 determine the "length" of the wave function.// |
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| | <del>We know from Linear Algebra that an //n// dimensional matrix //M// and the Eigenvalue/vector equation can be solved for <math>(n-1)</math> variables and <math>\lambda</math>. Multiplication by the matrix M represents a linear transformation of <math>\psi</math>, and the eigenvalue equation represents a transformation that maps all values of <math>\psi</math> to zero.</del> |
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| * For the Eigenvalue equation, the matrix //M// does //not// change the direction of the eigenvector <math>\psi</math>, only its magnitude; it is equivalent to multiplication by a //constant//. | For most physics applications, the matrix is Hermitian, and consequently, its eigenvectors are perpendicular, so //they usually form a orthonormal basis with which all other vectors can be expressed by their linear combinations.// A vector **x** can be <del>resolved</del> //decomposed// into its <del>perpendicular</del> components projected onto the eigenvectors quite easily. //Note that even if the eigenvectors are not orthogonal, as long as they are linearly **independent**, decomposition of vectors is possible, though figuring out the proper coefficients, c_n, will be trickier.// |
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| For most physics applications eigenvectors are perpendicular, so a vector **x** can be resolved into its perpendicular components projected onto the eigenvectors quite easily. | |
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| The Hydrogen Atom has an infinite number of Energy levels, so an infinite number of eigenvalues are possible. This also implies that the transformation matrix //M// can be infinite-dimensional. | The Hydrogen Atom has an infinite number of Energy levels, so an infinite number of eigenvalues are possible. This also implies that the transformation matrix //M// can be infinite-dimensional. |
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| * Every measurement of Energy will return the Exact same value, E. | * Every measurement of Energy will return the Exact same value, E. |
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| | * more on this topic on [[lec_notes_0916|tomorrow]]. |
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| **To go back to the lecture note list, click [[lec_notes]]**\\ | **To go back to the lecture note list, click [[lec_notes]]**\\ |