lethe, I read your post in the Quantum Physics thread and I'm interested to know more. So, I've taken the liberty to start a different thread as you suggested (SanDolphin61, I hope you don't mind) and ask a few more questions: 1) What is an eigenvalue and eigenvector? 2) Is it correct to say that a vector state only exist at sub-particle level? 3) If yes, then is it correct to say that at macro-level (our surrounding as our senses know it) all vector states have collasped to assume one eigenvector or another? 4) If yes, what then caused all these vector states to collaspe? Is it because at the macro-level, each vector state interacts with other vector states and thus, can be considered to have measured each other? My apologies if my questions do not seem to make any sense - I have a nagging feeling they don't. Anyway, it's close to midnight now. I look forward to your reply tomorrow morning.
Hi, I hope lethe won't mind me stepping in: "1) What is an eigenvalue and eigenvector?" An eigenvector v with eigenvalue <font face="symbol">l</font> of a matrix A satisfies the equation: A v = <font face="symbol">l</font> v (the matrix A working on v gives you a multiple of v as a results). This is a mathematical definition. In quantum mechanics, it happens to be that you can represent physically observable quantities by (to put it simply) a matrix. It turns out that the only allowed results for a measurement are the eigenvalues of that matrix... Example: You can represent the energy of a particle by a matrix H. When you calculate all possible eigenvalues of H you know all the allowed energy levels of that particle (this is a way to explain atomic spectra: an allowed energylevel is simply an eigenvalue of the energy-matrix). Eigenvectors have quite some interesting properties in quantummechanics. It turns out that if you do a measurement on a particle, that the wavefunction collapses to an eigenvector of the corresponding measurement matrix... An example: If you measure the energy, and you get E as a result (remember this is an eigenvalue for the energy-matrix, with some eigenvector belonging to it)... then the original wavefunction reduces to the eigenvector v that has E has an eigenvalue. "2) Is it correct to say that a vector state only exist at sub-particle level?" No, you can define a wavefunction (and its corresponding vectorstate) for an entire system of particles, even for macroscopic systems. It will be very complicated, and the tricky part is that it is (nearly) never decomposable into one-particle vector states. The entire set of particles is a lot more than the sum of the individual particles in quantum mechanics (but this is even true in classical mechanics, but that's a different story). Hope this helps a bit, Bye! Crisp
Eigenvalues (also called principal values or characteristic values) and eigenvectors have much wider applicability that quantum mechanics. Mathematically, an eigenvalue L is a value that satisfies the equ: [A]*[v] = L*[v] where [v] is any vector. Multiplication by the matrix [A], in general, causes rotation of the vector [v]. What the eigenvalue equation says is that there are certain coordinate systems in which the vector [v] isn't rotated - it is simply scaled up by an amount L. Examples are principal stresses (the eigenvalues of a stress tensor), natural frequencies of a vibrating system, and the buckling loads of a column under compression. Cheers, Ron.
