Lecture 2 - Statisticalinterpretationofthe wavefunction...

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Statistical interpretation of the wave function Quantum Mechanics and Probabilities
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Discrete random variables: probabilities Consider an event or an experiment who might have one of several (sometimes infinite but countable) outcomes. One can assign these outcomes some numerical values. The result is a random variable: a quantity that can take randomly one of many discrete values. Imagine that you repeat your experiment N times and record the number of times each value of the variable appears. You can frequency of each value i i N w N = If you repeat the series of experiments again, you will find that the frequencies have changed, but not too much. You can hypothesize that if you were to repeat the measurements very large number of times, the frequencies would take some permanent values. These values you would call probabilities lim i i N N p N →∞ = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 p 1 1 M i i p = = If total number of possible values of the variable is M
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Discrete random variables: moments Mean (expectation value) Variance Standard deviation (uncertainty all ii i xx p = () ( ) 2 2 2 2 all i x p x x Δ= = 2 all i x p
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Continuous random variables Since a continuous variable can take infinite number of values within any small interval, the probability of any particular number is zero. Thus, we define probabilities for continuous variables as probability to be within a specific interval of values. For small intervals this probability is proportional to the interval itself ( ) 0 00 0 0 0 () l i m x xx px x x x p x x ρ Δ→ = Δ +Δ < −Δ = Δ Δ = Δ Coefficient of proportionality is called probability density Normalization Mean (expectation value) Standard deviation (uncertainty x d x = 2 x x d x Δ= 1 xdx =
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Classical double slit experiment. Observed interference pattern is determined by intensity of the resultant wave ( ) ( ) ( ) () 12 ,c o s c o s 2c o s c o s 22 tA k R t A k R t kR R At ωω ω Ψ =− +− = −+ ⎛⎞ ⎜⎟ ⎝⎠ r 1 R 2 R ( ) 2 ,2 c o s 2 It A ∝Ψ = r maxima ( ) 2 n π = minima ( ) 1 n =+
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Quantum double slit experiment () ( ) 12 , ikR t t tA e A e ω −− Ψ= + r Each path through one of the slits represents a certain possible state of the quantum particle. Since the particle can arrive at a given point on the screen by one of two ways, the wave function corresponding to this state must be a superposition of two individual wave functions The brightness of the screen must be proportional to some real valued quantity related to the wave function. We introduce intensity of the wave function as ( ) ( ) 1 2 21 2 * ** 22 2 ,, , 11 2 c o s 2 t t t t ik R R ik R R It t t A e A e Ae Ae kR R Aee A ωω Ψ = ++ = ⎛⎞ + = ⎜⎟ ⎝⎠ rr r
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Statistical interpretation of the wave function The peculiarity of quantum double
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Lecture 2 - Statisticalinterpretationofthe wavefunction...

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