Download Commuting and Non-commuting Operators in Quantum Mechanics and more Study notes Physical Chemistry in PDF only on Docsity! 1 6-1 Commuting and noncommuting operators • In classical physics, there is no limit to what can be known about system (x, p, ) • In QM, two observables can be known simultaneously only if a measurement doesn’t change the state of the system. • This is only the case if is an eigenfunction of both operators • Two observables can be known simultaneously only if they have a common set of EF total kineticE , E ( )xψ 6-2 Consider operators • Measuring the observable corresponding to first and then that corresponding to equivalent to • Assume that is an EF of • Also true that System unchanged by measurement if EF of both operators! ˆ ˆ and A B  ( )n xψ B̂ ( )ˆˆ nB A xψ⎡ ⎤⎣ ⎦ ( ) ( ) ( )ˆˆ ˆn n n n n nB A x B x xψ α ψ β α ψ⎡ ⎤ = =⎣ ⎦ ( ) ( )ˆ ˆ n n n nA B x xψ α β ψ⎡ ⎤ =⎣ ⎦ ˆ ˆ and A B ( )n xψ 6-3 • In this case • Two operators that have a common set of EF are said to commute. • Test is if for arbitrary function f(x) operators commute. commutator • Example: Do commute? ( ) ( )ˆ ˆˆ ˆn nB A x A B xψ ψ⎡ ⎤ ⎡ ⎤= ⎣ ⎦⎣ ⎦ ( ) ( ) ( )ˆ ˆ ˆˆ ˆ ˆ, 0B Af x A B f x A B f x⎡ ⎤ ⎡ ⎤⎡ ⎤− ≡ =⎣ ⎦⎣ ⎦ ⎣ ⎦ [ ] ( ) ( ) ( )ˆ ˆ, x d xfdfx p f x i x i f x dx dx ⎛ ⎞ = − − = −⎜ ⎟ ⎝ ⎠ h h ˆ ˆ and xx p 2 6-4 Example: Do commute? Evaluate commutator = = 0 only for free particle ˆˆ andxp H ( ) ( ) ( ) ( ) 2 2 2 2 2 2 .2 2 d d d di V x f x V x i f x dx m dx m dx dx ⎛ ⎞ ⎛ ⎞⎛ ⎞− − + − − + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ h h h h ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d di V x f x i V x f x dx dx d d di V x f x i f x V x i V x f x dx dx dx di f x V x dx = − + = − − + = − h h h h h h ˆˆ ,xp H⎡ ⎤⎣ ⎦ 6-5 The Stern-Gerlach Experiment Silver atom beam in inhomogeneous field Atom deflection determined by z component of magnetic moment N S z x y 6-1 5 6-12 Create particle with limited certainty in momentum Do so by superposing plane waves Momentum somewhere in range ( ) 0 0( )1 1 , with 2 2 n m ik x i k n k x n m x Ae A e k kψ = + ∆ =− = + ∆ <<∑ 0 0 pk m k k m k− ∆ ≤ ≤ + ∆ h 6-13 What is P(x) for this wave function? Result is shown by red curve. -3 x10-10 -1 x10-10 1x10-10 3x10-10 Distance (m) 0 21 individual waves shown to right 6-14 Quantify ∆p and ∆x relationship Heisenberg uncertainty principle EP6.2: Assume that you could carry out the double slit experiment with electrons using a slit spacing of b = 10.0 nm. To see diffraction, we choose λ = b, and because diffraction requires reasonably monochromatic radiation, we choose Show that with these parameters, the uncertainty in the position of the electron is greater than the slit spacing b. 2 p x∆ ∆ ≥ h 0.01p p ∆ = 6 6-15 Using the de Broglie relation, is given by and The minimum uncertainty in position is which is greater than the slit spacing. This offers an explanation for the observation that the electron appears to go through both slits simultaneously! Trajectory not well defined for these parameters! 34 -26 -1 -10 6.626 x 10 J s 6.626 x 10 kg m s 100 x 10 m hp λ − = = = -28 -10.01 6.626 x 10 kg m sp p∆ = = 34 28 1 8 1.055 x 10 J s 2 p 2 x 6.626 x 10 kg m s 7.9 x 10 m=79nm x − − − − ∆ = = ∆ = h p 6-1 6-1
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