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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online