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xkcd xkcd.com Section 2 Recap ► ► ► Principle of Superposition: quantum states show interference and require both an amplitude and a phase for the parts Superposition applies in time as well as space For any observable, measured values come from a particular set of possibilities (sometimes quantised). Some states (eigenstates) always give a definite value (and therefore are mutually exclusive). Model as an orthonormal set of basis vectors. ► Model physical states as normalised vectors Can be expanded in terms of any convenient set of eigenstates. ► Measurements on systems in a definite quantum state (not an eigenstate) yield random results with definite probabilties for each. Represent the probabilities of modulus-squared of coordinates |ci|2 for the corresponding eigenstates in the eigenbasis of the observable. ► Some features of the mathematical formalism (e.g. overall phase of the state vector) don’t correspond to anything physical. Section 2 Recap ► Change with time is represented by a linear, unitary time evolution operator, U(t0,t) Unless interrupted by a measurement U I as the time interval t0−t 0. From U we derive Hamiltonian operator, H, and the (timedependent) Schrödinger Equation For a closed system U = exp[−iHt / ħ ] ► Measurements cause apparently discontinuous change in the state vector (“collapse of the wave function”). After an ideal measurement yielding result ai , state is in corresponding eigenstate |ai ► Best way of preparing systems in given quantum state is measurement + selection of required state. ► Hamiltonian for a particle in a field is H = −.B = − S.B