Transcript operators
Operators •A function is something that turns numbers into numbers f x sin kx •An operator is something that turns functions into functions d •Example: The derivative operator O= d O f (x ) = f (x ) = d sin (kx ) = k cos (kx ) dx dx dx •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 xopy (x)= xy (x) •x is an operator, sometimes written as xop or X •There are lots of other operators as well, like momentum p k i x pop i x 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 x P x x •It is denoted by x x x x xdx * x x x dx 2 •For any operator, we can similarly get an average measurement O * x O x dx x p * x pop x dx * x dx i x 0 x 4 Sample Problem A exp 12 Ax 2 A A particle is in the ground state of a harmonic oscillator. What is the expectation value of the operators x, x2, and p? x * x dx A e x 2 Ax2 * x dx A e 2 p * dx i x i Note: x2 x2 More on this later A e Note: Always use normalized wave functions for expectation values! x 0 xdx Ax2 x2 1 2 A 2 x dx Ax 2 2 Ax e n Ax 2 xe m Ax 2 2 dx 0 p 0 dx n 1 2 A n21 if n is even if n is odd 12 , 23 12 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: E H 2 p H V x 2m H •The expectation value of the Hamiltonian is the average value you would get if you measure the energy E H 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 2 op p 2m V xop 2 V xop 2 2m x 2 i H t 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? 30 x 5 Lx x 2 L 2 2 L pop d 2 V x dx E H * * 2 dx 2m 0 dx 2m L 2 2 L 30 2 d 15 2 2 2 Lx x Lx x dx Lx x 2 dx 5 2 5 2mL 0 dx mL 0 30 2 1 2 1 3 L 30 2 1 3 1 3 5 2 Lx 3 x 2 L 3L 5 2 5 0 mL mL mL2 •Compare to ground state: •Often gives excellent approximations 2 2 4.935 E 2 2 2mL mL 2 Tricks for Finding Expectation Values •We often want expectation values of x or x2 or p or p2 •If our wave function is real, p is trivial d p * dx i dx i d d 2 2 dx dx dx dx 2i 2i p 0 2 •To find p , we will use integration by parts d d 2 * 2 dx * dx dx i 2 p 2 2 p 2 2 d dx dx 2 d * d dx dx dx Uncertainty •Recall: x2 x2. Why? •The difference between these is a measure of how spread out the wave function is •Define the uncertainty in x: x 2 x x 2 4 6 46 26 25 2 2 2 2 2 2 •We can similarly define the uncertainty in any operator: p 2 p p 2 2 O 2 O 2 O 2 Heisenberg Uncertainty Principle x p 12 0 x 4 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 exp 12 Ax 2 m A A x 0 x2 1 2 A p 0 •Much of the work was done five slides ago •We even found p, but since is real, it is trivial anyway •Now work out p2: 2 p2 2 2 d dx dx 2 A x e 2 A 1 Ax exp Ax dx 2 A 2 Ax2 2 p2 dx 2 A p p 2 1 2 2 •Now get the uncertainties x x x 2 2 1 2A p xp 1 2 2 A2