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| classes:2009:fall:phys4101.001:lec_notes_0914 [2009/09/20 10:17] – yk | classes:2009:fall:phys4101.001:lec_notes_0914 [2009/09/20 10:23] (current) – yk |
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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> | <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 most physics applications 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. | 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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| 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]]**\\ |