# 18 - Physics 6210/Spring 2007/Lecture 18 Lecture 18...

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Physics 6210/Spring 2007/Lecture 18 Lecture 18 Relevant sections in text: § 2.3 Stationary states and classical mechanics Here we use the oscillator to illustrate a very key point about the relation between classical and quantum mechanics. The stationary states we just studied do not provide any explicitly dynamical behavior. This is not a speciﬁc feature of the oscillator, but a general feature of stationary states in quantum mechanics. This is, at ﬁrst sight, a little weird compared to classical mechanics. Think about it: the probability distributions for position and momentum are time inde- pendent in any state of deﬁnite energy. In classical mechanics the position and momentum (and the energy) can, at each moment of time, be determined with certainty; the values for the position and momentum oscillate in time in every state but the ground state.* In light of the incompatibility of the position, momentum and energy observables in the quantum description, one cannot directly compare the classical and quantum predictions in the excited states. The quantum predictions are purely statistical, involving repeated state preparations – states speciﬁed only by their energy – and measurements of various observables. If we want to compare the quantum and classical descriptions we need to ask the right questions of classical mechanics — this means statistical questions. Let us pause for a moment to expand on this. In classical mechanics every solution of the equations of motion is a state of deﬁnite energy (assuming energy is conserved). Fix an energy E . Let us ask the same question that we ask in quantum theory: Given the energy what is the probability for ﬁnding the particle at various locations? To answer this question we deﬁne the probability for ﬁnding the classical particle to be in the interval [ x, x + dx ] to be proportional to the time spent in that region. (The proportionality constant is used to normalize the probability distribution.) For a classical oscillator at the point x , a displacement dx takes place in the time dt where (exercise) dt = dx q 2 E m - ω

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18 - Physics 6210/Spring 2007/Lecture 18 Lecture 18...

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