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The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008-Oct-22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum mechanics is a different fundamental formalism, in which observables such as position and momentum are not real numbers but operators; consequently there are uncertainty relations, e.g. Δ x Δ p & ~ , which say that as some observables become more sharply defined, others become more uncertain. Experiments show that quantum mechanics, not classical mechanics, is the correct description of nature. Here is a summary of the essentials of quantum mechanics, focussing on the case of a single non-relativistic particle (eg an electron) in one dimension. This is not a complete review of everything you need to know: it is a quick outline of the basics to help you get oriented with this challenging subject. 1. States . The state of the system is given by a wavefunction ψ ( x ). The wavefunction is complex, and gives the amplitude for finding the particle at position x . The probability density is the square modulus of the amplitude, so in one dimension the probability to find the particle located between x and x + dx is P ( x ) dx = | ψ ( x ) | 2 dx . (1) This means that if you start off with an “ensemble” of identical copies of the system, all with the same wavefunction ψ ( x ), and in each member you measure the position of the particle, then you will get different results from different members of the ensemble. The probability of getting particular answers is given by (1). A physical state must be normalized, so that all the probabilities add up to 1: Z | ψ ( x ) | 2 dx = 1 . (2) However, for some purposes we will find it convenient to use unnormalized states. 2. Operators . Each observable corresponds to a linear operator. A linear operator is something that acts on a state and gives another state, ie it changes one function into another. position operator ˆ x defined by ˆ ( x ) = ( x ) momentum operator ˆ p defined by ˆ ( x ) = - i ~ ∂ψ ∂x (3) 1
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For example, when the position operator acts on the state ψ ( x ) = 1 / ( a 2 + x 2 ), it gives x/ ( a 2 + x 2 ), while the momentum operator gives 2 i ~ x/ ( a 2 + x 2 ) 2 .
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