Lect09 - Labs will be in room 257 (2nd floor) this week....

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Labs will be in room 257 (2 nd floor) this week. Need AD sign-on. See supplementary notes at end of this lecture to try to get big picture for QM See website for link to new formula sheet, through this week. http://www.falstad.com/qm1d/
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Superposition & Time-Dependent Quantum States (x,t) | 2 U= x U= 0 x L
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Overview Overview z Superposition of states and particle motion z ‘Packet States’ in a Box z Measurement in quantum physics z Schrödinger’s Cat z Time-Energy Uncertainty Principle
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Time Time - - independent independent SEQ SEQ z Up to now, we have considered quantum particles in stationary states ,” and have ignored their time dependence Remember that these special states were associated with a single energy (from solution to the SEQ) “eigenstates” of energy ) ( ) ( ) ( ) ( 2 2 2 2 x E x x U dx x d m ψ = + = U= ψ (x) 0 L U= n=1 n=2 x n=3 “Functions that fit”: ( λ = 2L/n) ψ (x) 0 L x “Doesn’t fit”:
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Superposition: a roadmap Superposition: a roadmap z The fundamental (time-dependent) SEQ is LINEAR so its solutions obey superposition. z There is NO RULE forbidding states with a range of different E, just as there was no rule forbidding states with ranges of x or p. z E.g. Superpositions of the n=1 and n=2 states of particle in box z The states of fixed E did nothing! They just sat there. http://www.falstad.com/qm1d/ z (actually just rotated in complex plane- no change in probability density) z All interesting events come from states with more than one E, superpositions of the single-E states we described. z So now we have to learn how to treat those mathematically. (x,t) | 2 U= U= 0 x L
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Review Review Complex Numbers The equation, e i θ = cos θ + isin θ , might be new to you. It is a convenient way to represent complex numbers. It also (once you are used to it) makes trigonometry simpler. a) . a) b) a) The Argand diagram of a complex number, A , puts Re(A) on the x -axis and Im(A) on the y - axis. Draw an Argand diagram of e i θ Notice the trig relation between the x and y components. θ is the angle of A from the real axis. In an Argand diagram, e i θ looks like a vector of length 1, and components (cos θ , sin θ ) . b) Suppose that q varies with time, q = wt. How does the Argand diagram behave? At t = 0 , θ = 0 , so A = 1 (no imaginary component). As time progresses, A rotates counterclockwise with angular frequency ω . Re(A) Im(A) θ A = e i θ Re(A) Im(A) θ = ω t A = e i ω t t = 0 ce i θ ( c and θ both real), is a complex number of magnitude, |c| . The magnitude of a complex number, A , is |A| = (A*A) , where A* is the complex conjugate of A : Im(A*) = -Im(A).
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Lecture 9, Act i Lecture 9, Act i We know that 1. What is (-i)i? a. –i b. -1 c. +1 2. What is 1/i? a. –1 b. -i c. +i 3. What is |e i φ | 2 ? a. 0 b. e 2i φ c. 1 1 i
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Lecture 9, Act i Lecture 9, Act i 1 i We know that 1. What is (-i)i?
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This note was uploaded on 09/15/2008 for the course PHYS 214 taught by Professor Debevec during the Spring '07 term at University of Illinois at Urbana–Champaign.

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Lect09 - Labs will be in room 257 (2nd floor) this week....

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