Lecture4 - PHYS 360 Quantum Mechanics Wed Jan 19, 2011...

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Unformatted text preview: PHYS 360 Quantum Mechanics Wed Jan 19, 2011 Lecture 4: Since QM relies on probabiliDes, how do we quanDfy uncertainDes? Recap of last lecture…. Vibra&ng String Schrödinger (QM) Wave equaDon ∂ 2 y( x, t ) 1 ∂ 2 y( x, t ) ∂Ψ ( x , t ) 2 ∂2 Ψ( x, t ) =2 i =– + V Ψ( x, t ) 2 ∂x c ∂t 2 ∂t 2m ∂x 2 SeparaDon of variables Eigenvalue equaDon EigenfuncDon example: “Eigenstate” Wave funcDon yn ( x, t ) = fn ( x )gn (t ) Ψ n ( x, t ) = ψ n ( x )ϕ n (t ) 2 2 d ψ n ( x) − + V ψ n ( x ) = Enψ n ( x ) 2 2 m dx d 2 f (x) = − k 2 f (x) dx 2 Length L, ends clamped: Infinite square well, width “a”: fn ( x ) = an sin( nπ x ) L ψ n ( x ) = An sin( nπ x ) a yn ( x, t ) = fn ( x ) exp(−iω nt ) Ψ n ( x, t ) = ψ n ( x ) exp(−iω nt ) ∞ y( x, t ) = nπ x ∑ an sin( L ) exp(−iω nt ) n =1 ∞ Ψ ( x, t ) = ∑ An sin( n =1 nπ x ) exp(−iω nt ) a Re(Ψ ( x, t )) Ψ ( x, t ) 2 n=2 0.0 0.5 1.0 0.0 0.5 1.0 n=3 0.0 0.5 1.0 0.0 0.5 1.0 Ψ ( x, t ) = Ψ 2 ( x, t ) + Ψ 3 ( x, t ) 2π x 3π x = sin( ) exp(iω 2t ) + sin( ) exp(iω 3t ) a a 0.0 0.5 1.0 Not staDonary! ∫ a b Ψ ( x, t ) dx = { Probability of finding the parDcle between a and b, at Dme t. 2 } This curve can be mapped out by making a measurement on many different copies of idenDcally prepared systems (an ensemble). What happens to one system a^er you make a single measurement? ∫ a b Ψ ( x, t ) dx = { Probability of finding the parDcle between a and b, at Dme t. 2 } The wave funcDon “collapses” so that a follow‐up measurement sDll finds the parDcle at C. Did the measurement change the system? Did the measurement “cause” the parDcle to be at C? ∫ a b Ψ ( x, t ) dx = { Probability of finding the parDcle between a and b, at Dme t. 2 } Where was the parDcle before the measurement found it at C? Was it already at C? (Realist answer) Was it not at a definite posiDon? (Orthodox answer) Is the quesDon invalid? (AgnosDc answer) Can this be answered by experiment? YES! (…realists are wrong!) Probability – Discrete Variables Imagine a room containing 14 people, whose ages are: One person aged 14, one person aged 15, three people aged 16, two people aged 22, two people aged 24, and five people aged 25. OR, Imagine making 14 measurements of the vibraDonal energy level of a sample of diatomic molecules (which are not in thermal equilibrium). The energy follows the equaDon E ‐ Eo=nħω. The results are: Once n=14, n=15 Three Dmes n=16 Twice n=22, n=24 Five Dmes n=25 Let N(j) be the number with value j: N (14 ) = 1, N (15 ) = 1, N (16 ) = 3, N (22 ) = 2, N (24 ) = 2, N (25 ) = 5 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 The total number of measurements (people) is: The probability that the result of any measurement is 15 is: P(15) = N= Σ0 N ( j ) = 14. j= ∞ N (15) 1 =. N 14 In general, N ( j) P( j ) = . N Also note: Σ P( j ) = 1. j =0 ∞ 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Most probable result? 25 Median value: 23 (half are larger, half are smaller) Mean value? j ΣjN = N ( j) = Σ0 j P ( j ). j= ∞ 14 + 15 + 3 × 16 + 2 × 22 + 2 × 24 + 5 × 25 294 = = 21 14 14 Note: None of the measurements yielded 21, the average value. 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 What is the average of the square of the values? j 2 = Σ j 2 P( j ) j =0 ∞ The average value of any funcDon of j is given by: f ( j) = Σ f ( j ) P(j ). j =0 ∞ Note: the probability distribuDon P(j) serves as a weighDng funcDon. Let’s jump ahead to Quantum Mechanics and conDnuous variables. The weighDng funcDon is the square of the wave funcDon: ∫ a b Ψ ( x, t ) dx = { Probability of finding the parDcle between a and b, at Dme t. 2 } Discrete f ( j) = f ( x) = Σ f ( j ) P(j ). j =0 ∞ ConDnuous +∞ −∞ ∫ f ( x )ψ * ( x )ψ ( x ) dx 9 8 7 6 5 4 3 2 1 0 σ 0.45 Median j=5 Mean <j>=5 Most probable j=5 9 8 7 6 5 4 3 2 1 0 1 σ 2.45 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 How do we characterize how “localized” a distribuDon is? deviation from average Δj = j − j However, Δj = ∑ ( j − j )P( j ) = ∑ jP( j ) − j P( j ) = j − j (1) = 0 2 σ2 ≡ ( Δj )2 = ...... = j 2 − j 2 = variance Instead, σ= j 2 −j = the standard deviation Why are we so concerned with the standard deviaDon? When we discuss the Heisenberg uncertainty principle, instead of a qualitaDve statement like: Δx Δp > 2 …we will need a quanDtaDve statement like: σ xσ p ≥ 2 2 where σ x = x 2 − x , and 2 x = ∫ xΨ * ( x, t )Ψ ( x, t ) dx ...
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This note was uploaded on 04/23/2011 for the course PHYS 360 taught by Professor Durbin,stephen during the Spring '11 term at Purdue.

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