5.61 Fall 2007 Lecture #8 page 1 QUANTUM MECHANICAL PARTICLE IN A BOX Summary so far: ∞ ∞ V( x ) 0 0 ( ) V (0 ≤ x ≤ a ) = 0 V x < 0, x > a = ∞ n2 h2 En = 8ma 2 a x k= ( ) ⎛ nπ x ⎞ ψ ( 0 ≤ x ≤ a ) = B sin ⎜ ⎝ a ⎟⎠ ψ x < 0, x > a = 0 n nπ a λ= 2a n n = 1,2,3,... () What is the “wavefunction” ψ x ? Max Born interpretation: () ψ x 2 () () =ψ * x ψ x is a probability distribution or probability density for the particle ∴ () ψ x 2 dx is the probability of finding the particle in the interval between x and x + dx This is a profound change in the way we view nature!! We can only know the probability of the result of a measurement – we can’t always know it with certainty! Makes us re-think what is “deterministic” in nature. Easy implication: ⇒ ∫ x2 x1 () Normalization of the wavefunction 2 ψ x dx = probability of finding particle in interval xxx≤≤ 12 The total probability of finding the particle somewhere must be 1. For a single particle in a box, 5.61 Fall 2007 Lecture #8 ∫ ∫ ∞ −∞ a 0 () 2 ψ x dx = ∫ ( ) a 0 2 ψ x dx = 1 ⎛ nπ x ⎞ B 2 sin 2 ⎜ dx = 1 ⎝ a ⎟⎠ () ψn x = ⎛ nπ x ⎞ 2 sin ⎜ a ⎝ a ⎟⎠ 2 ⇒ n = 1,2,3,... ( ) 2 ( ) ( ) ( ) ( ) ( ) 2 = 2 a sin 3π x a 2 ψ 2 = 2 a sin 2 2π x a 2 ψ 1 = 2 a sin 2 π x a 2 a Normalized wavefunction () ) ( ) ( 3 ()() 2232sin3axaψπ= ψ Normalization condition B= ψ 4 = 2 a sin 4π x a 2 page 2 Interpretation of ψ x based on measurement Each measurement of the position gives one result. Many measurements give a probability distribution of outcomes. Expectation values or average values For a discrete probability distribution 0.4 0.35 0.3 0.25 e.g. 0.2 P 0.15 0.1 0.05 0 2 4 2 6 8 x 10 12 14 5.61 Fall 2007 Lecture #8 page 3 = average value of x = 4(0.1) + 6(0.1) + 8(0.2) + 10(0.4) + 12(0.2) = 4(P4) + 6(P6) + 8(P8) + 10(P10) + 12(P12) where Px is probability that measurement yields value “x ” x = ∑ xPx ⇒ Now switch to continuous probability distribution () Px → ψ x ∑ → ∫ x = ⇒ Similarly x2 = ∫ ∞ −∞ ∫ ∞ −∞ 2 dx () 2 x ψ x dx () 2 x 2 ψ x dx Often written in “sandwich” form () () = ∫ ψ * ( x ) x ψ ( x )dx x = x2 ∫ ∞ −∞ ψ * x xψ x dx ∞ 2 −∞ For particle in a box x = ⎛ nπ x ⎞ 2 a x sin 2 ⎜ dx ∫ a 0 ⎝ a ⎟⎠ 5.61 Fall 2007 Integrate by parts Lecture #8 → x = a 2 The average particle position is in the middle of the box. page 4