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Quantum states and operators_PHYS3316

Quantum states and operators_PHYS3316 - Quantum States and...

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Quantum States and Operators Physics 3316 - Basics of Quantum Mechanics Spring 2010 Quantum States The first step in building a theory capable of predicting the results of experiments carried out on a physical system is to develop a description of the different possible physical states of the system and a method for the prediction of the outcome of experiments carried out on a system in a given state. Only once this is done do we then go about the business of building a theory to give us the possible states of a system and to describe how a system moves from state to state as a function of external and internal influences. 1 States and Observables In classical mechanics, the state of a system may be specified by giving the coordinates { vectorx i } and velocities { vectorv i } of all of the particles in the system. Once this is done, all measurable quantities such as the momenta { vector p i } , total energy H = p 2 i 2 m + V ( { x i } ) and angular momentum vector L = vector r i × vector p i of the system may be determined. Such measurable quantities are known as observables . Under the classical principle of determinism, once the state of the system is specified at time t = 0, the laws of motion may be solved to determine the state of the system { vectorx i ( t ) ,vectorv i ( t ) } for all later times. The situation in quantum mechanics is not much different. We may still make measurements of the familiar physical observables mentioned above (and some new ones such as spin or polarization which we will learn about later). As we have seen in our discussions of the uncertainty principle and interference experiments, although the results of individual measurements are unpredictable on the quantum scale, measurements on a system in a given state yield a well-defined distribution of results. If we simply accept this notion and generalize our concept of “experiment” to the procedure of determining the distribution associated with many measurements of the same observable over a set of identically prepared systems, then we may maintain the idea that the quantum state of a system determines the results of all experiments measuring physical observables. As we shall refer to the quantum state of a system often, we shall make a special symbol for it, | ϕ ) . The symbol |) is called a “ket.” We may place different symbols inside the ket to distinguish different states. For instance, our system may be in the state | ψ ) at time t = 0 and in the state | φ ) at some later time. Keep in mind that, in general, | ψ ) does not determine the result of any individual measurement of an observable, only probability distributions such as P ( x ) dx , the probability of finding a particle with position x in the range ( x x + dx ), or ˜ P ( p ) dp , the probability of finding the momentum of the particle in the range ( p p + dp ). In the case of photons we may also consider the observable of spin or polarization and measure P ( σ ), the probability of observing a right ( σ = 1) or left ( σ = 1) circularly polarized photon.
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