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Unformatted text preview: Lecture Notes on Hilbert spaces State State of a physical system at a given time is basically all information that identifies the particular state the system is in. For example, “I am reading these notes” is a description of your current state, but not a complete one. To describe your state accurately, you also need to give the position of your hands, head, etc., but should also give the position and velocity of each particle your body contains. The state of a classical particle in 1D can be specified completely by giving its position x and momentum p (at a given time). By using this information, you can compute everything that can be computed. For example, you can find its energy, its state one second later (i.e., position and momentum 1 s later) as a function of the current state ( x,p ). The state for such a particle can be mathematically represented by a point in the 2-dimensional phase space . For the general case of a classical system composed of N classical particles moving in 3D, the state can be specified by giving a total of 6 N coordinates of position and momenta. Equivalently, the state of that system at a given time, can be mathematically represented by a point in 6 N-dimensional phase space. For quantum mechanical systems, you already know that the classical phase space is not suitable for describing any state of the system. There is a more complicated (but simpler in certain respects) mathematical structure for representing the states of quantum mechanical systems. This has to be so, because new concepts not met in classical systems, such as super- position , arise in quantum mechanics and your mathematical apparatus should be appropriate to handle these. It appears that, any state of a quantum mechanical system can be mathematically rep- resented as a normalized vector in a certain Hilbert space . The classical phase space then has to be replaced by this Hilbert space. Such vectors satisfy the basic property you expect from states, i.e., they contain all the information about the actual physical state the system is in. Any property of the system in that particular state can be computed by using that vector. But it appears that, some certain properties familiar from classical mechanics are not meaningful and hence cannot be computed. For example, you cannot get a meaningful answer to “what is the exact position of this particle in this particular state” and you cannot compute it. But you can ask “what is the probability that this particle will appear around x when I measure its position” and you can get the answer by computing a certain expression involving the vector that contains the state information. In particular, the vector contains all information that can be produced in actual experiments....
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This note was uploaded on 02/11/2012 for the course MATH 435 taught by Professor Starg during the Spring '11 term at Al Ahliyya Amman University.
- Spring '11