![]() ![]() This is a fundamental postulate of quantum mechanics. Then the probability to find the Lambda i is, I'm not writing Lambda i equal to this one, the probability to find Lambda i equals to Ci absolute value squared. Here I'm assuming the wave function is normalized, integrated everywhere, the probability to find the particle is one. So if the wave function is normalized, you have just equal to this Ci absolute value squared, otherwise it's proportional. So the probability to find the observable, which is the eigenvalue Lambda i corresponding to this Ci absolute value squared. So what's the simplest way to translate it into a probability? That the probability to find out the eigenvalue Lambda i, how does Lambda i arises? Because the eigenstate is the corresponding eigenstate having eigenvalue Lambda i. Ci should be some positively correlated to the probability to find the state. If you do a measurement, if the Ci is larger, this state is more important, then you have a larger probability to find out this state in this Psi Lambda i. Ci in general could be a complex number, but if the magnitude of Ci is larger, that corresponds to this eigenstate, Psi Lambda i, the state with eigenvalue Lambda i, that eigenstate is more important in composing, in taking part of this Psi x. The eigenfunction here has the weight which is Ci contributing to Psi. Here very intuitively, this Ci is a weight for the superposition. Now, this is the case for discrete state. ![]() Later we will come to continuous case, which is more similar to momentum and position. ![]() It could be in superposition of that plane wave, and the superposition coefficients are Ci for discrete case. This Psi is the system that it could be in any state. For example, if you want to measure momentum, the eigenstate are just the plane wave. Eigenstate is determined by what you want to measure. But the eigenstate is determined by operator. So given different systems in general, then you will have different Cis. Whatever system that you have, Psi can be decomposed into a summation of Ci and Psi Lambda i of x, where this Psi Lambda i is the eigenstate with eigenvalue Lambda i, and then the corresponding coefficient is Ci. First, if Lambda is discrete, then a general state can be decomposed into the superposition of these discrete states, such as a general wave function. Now in general, what about a general state? To talk about the general state, we have to consider two possible cases, actually they're pretty unified, one case is the possible outcome of the measurement is discrete and another is continuous. In the previous video, we have shown that if we do a measurement on a state which happens to have a definite value of this observable, of this measurement, then we can use the eigen equation to describe this measurement process. ![]()
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