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Published byGarry Barnett Modified over 9 years ago
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Operators A function is something that turns numbers into numbers An operator is something that turns functions into functions Example: The derivative operator In quantum mechanics, x cannot be the position of a particle Particles don’t have a definite position Instead, think of x as something you multiply a wave function by to get a new wave function x is an operator, sometimes written as x op or X There are lots of other operators as well, like momentum
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Expectation Values Suppose we know the wave function (x) and we measure x. What answer will we get? We only know probability of getting different values Let’s find the average value you get Recall | (x)| 2 tells you the probability density that it is at x We want an expectation value It is denoted by x For any operator, we can similarly get an average measurement
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Sample Problem A particle is in the ground state of a harmonic oscillator. What is the expectation value of the operators x, x 2, and p? Note: x 2 x 2 More on this later Note: Always use normalized wave functions for expectation values!
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The Hamiltonian Operator In classical mechanics, the Hamiltonian is the formula for energy in terms of the position x and momentum p In quantum, the formula is the same, but x and p are reinterpreted as operators Schrodinger’s equations rewritten with the Hamiltonian: Advanced Physics: The Hamiltonian becomes much more complicated More dimensions, Multiple particles, Special Relativity But Schrodinger’s Equations in terms of H remain the same The expectation value of the Hamiltonian is the average value you would get if you measure the energy
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Sample Problem A particle is trapped in a 1D infinite square well 0 < x < L with wave function given at right. If we measure the energy, what is the average value we would get? Compare to ground state: Often gives excellent approximations
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Tricks for Finding Expectation Values We often want expectation values of x or x 2 or p or p 2 If our wave function is real, p is trivial To find p 2, we will use integration by parts
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Recall: x 2 x 2. Why? The difference between these is a measure of how spread out the wave function is Define the uncertainty in x: Uncertainty We can similarly define the uncertainty in any operator: Heisenberg Uncertainty Principle
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Sample Problem A particle is in the ground state of a harmonic oscillator. Find the uncertainty in x and p, and check that it obeys uncertainty principle Much of the work was done five slides ago We even found p , but since is real, it is trivial anyway Now work out p 2 : Now get the uncertainties
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