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ideasOfQM - Physics 195a Course Notes Ideas of Quantum...

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Physics 195a Course Notes Ideas of Quantum Mechanics 021024 F. Porter 1 Introduction This note summarizes and examines the foundations of quantum mechanics, including the mathematical background. 2 General Review of the Ideas of Quantum Mechanics 2.1 States We have in mind that there is a “system”, which is describable in terms of possible “states”. A system could be something simple, such as a single electron, or complex, such as a table. Suppose we have a system consisting of N spinless particles. We use the term “particle” to denote any object for which any internal structure is unimportant. Classically, we may describe the state of this system by specifying, at some time t the generalized coordinates and momenta: { q i ( t ) , p i ( t ) , i = 1 , 2 , . . . , N } , (1) where the spatial dimensionality of the q i and p i is implicit. The time evolu- tion of this system is given by Hamilton’s equations: ˙ q i = ∂H ∂p i (2) ˙ p i = ∂H ∂q i (3) In quantum mechanics, it is not possible to give such a complete speci- fication to arbitrary precision. For example, the limit to how well we may specify the position and momentum of a particle in one dimension is limited by the “uncertainty principle”: ∆ x p 1 / 2, where ∆ indicates a range of possible values. We’ll investigate this relation more explicitly later, but for 1
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now it should just be a reminder of your elementary quantum mechanics un- derstanding. We must be content with selecting a suitable set of quantities which can be simultaneously specified to describe the state. We refer to this set as a “Complete Set of Commuting Observables” (CSCO). Specifying a CSCO corresponds to specifying the eigenvalues of an ap- propriate complete set of commuting Hermitian operators, for the state in question. Measurements (eigenvalues of Hermitian operators) of other quan- tities cannot be predicted with certainty, only probabilities of outcomes can be given. The evolution in time of the system is described by a “wave equa- tion”, for example, the Schr¨ odinger equation. 2.2 Probability Amplitudes The quantum mechanical state of a system is described in terms of waves, called probability amplitudes , or just “amplitudes” for short. Note that probabilities themselves are always non-negative, so it is more difficult to imagine the probabilities themselves as wavelike. Instead, the probabilities are obtained by squaring the amplitudes: Probability ∼ | ψ | 2 , (4) where ψ stands for the amplitude. More explicitly, the probability of ob- serving state variable ( e.g. , position) x in volume element d 3 ( x ) around x is equal to: | ψ ( x ) | 2 d 3 ( x ) . (5) A quantum mechanical probability is analogous to the intensity of a classical wave. The quantum mechanical wave evolves in time according to a time evo- lution operator, e iHt involving the Hamiltonian, H . Hence, if e iHt ψ 0 ( x ) = ψ ( x , t ) , (6) where ψ 0 ( x ) is the wave function at t = 0 in terms of coordinate position x , then differentiation gives: i ( x , t ) dt = ( x , t ) . (7) We recognize this as the Schr¨odinger equation. Thus, the temporal frequency of the wave is determined by the energy structure. For a particle of energy
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