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classes:2009:fall:phys4101.001:lec_notes_1016

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classes:2009:fall:phys4101.001:lec_notes_1016 [2009/10/17 23:12] – x500_nikif002classes:2009:fall:phys4101.001:lec_notes_1016 [2009/10/17 23:50] (current) – x500_nikif002
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 <math>\ f(x) = \sum_n c_n\psi_n</math> can be written as <math>\ f(x) <=>\normalsize \left(\large\begin{array}{GC+23} \\c_{1}\\c_{2}\\c_{n}\end{array}\right)\ {\Large</math> <math>\ f(x) = \sum_n c_n\psi_n</math> can be written as <math>\ f(x) <=>\normalsize \left(\large\begin{array}{GC+23} \\c_{1}\\c_{2}\\c_{n}\end{array}\right)\ {\Large</math>
 where <math>\ c_n = \int f(x)*\psi_n(x) \,\mathrm dx</math>  where <math>\ c_n = \int f(x)*\psi_n(x) \,\mathrm dx</math> 
-  *//the reason we can do this is because all stationary states are orthonormal//+  *//the reason we can do this is because all stationary states are orthonormal. The stationary states in function space are like the basis vectors in //<math>R^n</math>
   *the above vector can have finite or infinite number of components depending on how many stationary states there are.   *the above vector can have finite or infinite number of components depending on how many stationary states there are.
   *in the language of vectors, operators are transformations and are featured by Matrices. (I think one point of the lecture was the following: by operation on a function you get another function analogous to for example when you apply addition rules on a vector you get another vector which belongs to the original vector space)   *in the language of vectors, operators are transformations and are featured by Matrices. (I think one point of the lecture was the following: by operation on a function you get another function analogous to for example when you apply addition rules on a vector you get another vector which belongs to the original vector space)
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 <math>\ <\psi|Q\psi>  = <Q\psi|\psi>* ,</math> and if it is real we are not changing anything by changing the order. //This is an easy way to see that a real expectation value is necessary and sufficient to show that the operator is Hermitian.// <math>\ <\psi|Q\psi>  = <Q\psi|\psi>* ,</math> and if it is real we are not changing anything by changing the order. //This is an easy way to see that a real expectation value is necessary and sufficient to show that the operator is Hermitian.//
   *determinate state is a state with no uncertainty and each measurement of some quantity Q gives back the same result q. //In other words, the standard deviation of the observable is 0.//   *determinate state is a state with no uncertainty and each measurement of some quantity Q gives back the same result q. //In other words, the standard deviation of the observable is 0.//
-  *//This is something that confused me. Which operators can we define an eigenvalue for? All linear operators? All Hermitian operators? Certainly we can define some non-linear operators that don't have eigenvalues (such as the squaring operator). Looking at the book, it explains it. Determinate states for an operator Q are states that are eigenfunctions of Q. So //<math>\hat{Q}f(x)=q f(x)</math>//, where q is a constant. So, as far as I understand, this means that, for example, all solutions to the Schrodinger equation are determinate states of the Hamiltonian operator.//+  *//This is something that confused me. Which operators can we define an eigenvalue for? All linear operators? All Hermitian operators? Certainly we can define some non-linear operators that don't have eigenvalues (such as the squaring operator). Looking at the book, it explains it. Determinate states for an operator Q are states that are eigenfunctions of Q. So //<math>\hat{Q}f(x)=q f(x)</math>//, where q is a constant. So we can define eigenvalues for an operator Q if the function it is operating on is a determinate state of Q. As far as I understand, this means that, for example, all solutions to the Schrodinger equation are determinate states of the Hamiltonian operator.//
   *//Another new piece of notation: //<math>|q_n></math> //indicates the wavefunction corresponding to eigenvalue //<math>q_n</math>//. Such that //<math>\hat{Q} |q_n>=q_n|q_n></math>   *//Another new piece of notation: //<math>|q_n></math> //indicates the wavefunction corresponding to eigenvalue //<math>q_n</math>//. Such that //<math>\hat{Q} |q_n>=q_n|q_n></math>
  
classes/2009/fall/phys4101.001/lec_notes_1016.1255839124.txt.gz · Last modified: 2009/10/17 23:12 by x500_nikif002

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