Transcript Quantum
Quantum New way of looking at our world Classical vs Quantum Typically a student develops an intuition about how the world works using classical mechanics and practical experience. Newton’s laws Objects dropping There are some necessary realignments but students tend to accept much of what is explained and integrate it into there world model Quantum mechanics requires a completely new formulation. The connections with practical experience are more difficult but not impossible. There is a tendency to be much more demanding at this stage. Students are less likely to believe the principles and ideas. QM Require a CHANGE in Perception State of the system | STATE > Labeled by the observables | mass, momentum, energy > Allows us to predict the results of measurements STATE Classical system | mass,x,y,z,vx,vy,vz, Q > Given a particle in a potential well there is an infinite set of possible orbits. The mass and charge of course pick certain orbits by choosing the strength of the potential and of course the inertial response of the particle. The position and velocity specify which of these classical orbits the particle will follow. Thus given the value of these variables and of course the forces (potential), the orbit is determined and the location and velocity at any time can be found. Our states will be elements of a vector space. That means several important things. Most of the properties of vector spaces can be understood by considering the well know space of 3-vectors. There is a mathematical process called the dot product. Vectors have components that determine (STRENGTH) the amount in any direction. Vectors can be expressed in terms of a basis. Basis vectors need to be complete. You need to pick basis vectors that allow you to specify all directions. Usually this is x,y,z. There are infinite possibilities for the basis vectors. You can create mathematical relationships that define how the components change. (Transformations & Representations). Quantum variation For every observable we define an operator. Aˆ | a a | a The state is specified by a value of the observable a. **********Eigenvector-Eigenvalue************** True for only a subset of all the states. More on operators The most general way that an operator can act on a state is to transform it to a new state. Aˆ | | Not all operators represent observables of the system. Rotation is an OPERATION Pˆ momentum xˆ position Hˆ energy Lˆ angular momentum Sˆ spin Jˆ Total angular momentum iHˆ t ˆ U time evolution e iJˆ ˆ R rotation e | i | i Dot Product | Basis vectors | i | i |i | i | i | i | i For now we note [ xˆ, Pˆ ] i h is the commutator for position and momentum. This equation determines the relationship between the observables and makes position and momentum mutually exclusive in the sense that both cannot be simultaneously precisely measured for a quantum system. Abner Shimony 1 State “Associated with every physical system is a complex linear vector space V, such that each vector of unit length represents a state of the system. “ Abner Shimony 2 Observables “There is a one to one correspondence between the set of eventualities (observables) concerning the system and the set of subspaces of the vector space associate with the system, such that if e is an eventuality (observable) and E Is the subspace that corresponds to it, then e is true in a state |S> if and only if any vector that represents S belong to E; and is false in the state S if and only if any vector that represents S belongs to E(orthogonal) .” A states described by a vector that has components in both subspaces represents a state with an unspecified value for this observable e. Abner Shimony 3- Measurement “If |S> is a state and e is an eventuality (observable) corresponding to the subspace E, then the probability that e will turn out to be true if the initially the system is in state |S> and an operation is performed to actualize (measure) it. probS (e) || PE v || 2 v is a unit vector representing a state with an observable value e … and PE is the projection.” Abner Shimony 4 composites i.e. atom e,P “If 1 and 2 are two physical systems, with which the vector spaces V1 and V2 are associated, then the composite system 1 + 2 consisting of 1 and 2 is associated with the tensor product V1 x V2.” Abner Shimony 5 Time evolution “If a system is in a nonreactive environment between 0 -> t, then there is a linear operator U(t) such that U(t) |v> represents the state of the system at time t if |v> represents the state of the system at time 0. Furthermore, ||U(t) v||2 = || v||2 for all v in the vector space.”