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Quantum Physics 2005 Notes-3 Observables – (Chapter 5) Notes 3 Quantum Physics F2005 1 Introduction • Observables are physical attributes of a system that can be measured in the laboratory. • In quantum physics, in the absence of a measurement, a microscopic system does not necessarily have values of its physical properties. (A particle does not “have” a position until we measure it. It has a set of possible positions.) • We want to find out how to calculate observables from wavefunctions. The mathematical approach is through the introduction of operators. Notes 3 Quantum Physics F2005 2 • In quantum scale measurements, the measurement itself affects the state of the particle, so we cannot use classical definitions of observables. • Example: position and momentum. – The momentum is measured by measuring velocity. This involves the measurement of position at two times. The first measurement affects the momentum and hence the later position and consequent velocity. Notes 3 Quantum Physics F2005 3 Two ways to compute momentum 1) Given ! ( x, t ), use the wave function to compute " p ( p). Compute p from ∫ "* p"dp. 2) Find a way to express p as a function of x. Compute p from ∫ ! * p( x)!dx. Let’s compute each way for the lowest state of the square well. Notes 3 Quantum Physics F2005 4 Momentum in the square well ! ( x, t ) = 2 $ x i#t cos e L L 1 % 1 L/2 $ x ipx / & ipx / "( p) = ! = ( x , 0) e dx cos ∫ ∫ e dx L$ h & L / 2 2$ h &% L 1 L/2 $x = ∫ cos ( cos( px / h) + i sin( px / h) ) dx L$ h & L / 2 L = L/2 $ p px $ p 1 L/2 1 $x = & + dx x cos cos( ) cos cos ∫ ∫ L + h x dx L L h h & L / 2 L$ h & L / 2 L $ h 2 L/2 $ p $ p sin sin x & L + h x 1 L h = + $ p 2 L$ h $ p & L + h L h &L/ 2 = Notes 3 $ p L $ p L sin sin & L + h 2 1 L h 2 + $ p L$ h $ p & L + h L h Quantum Physics F2005 5 Momentum in square well L "( p) = 2 $ p L $ p L sin L & h 2 sin L + h 2 1 + $ p L L$ h $ p L & L + h 2 L h 2 $ L $ L p sin 2 & 2h p sin + L 2 2h "( p) = + 4$ h $ L $ L p p & + 2 2h 2 2h L for plotting, let's set = 2 (in some units). 2h Notes 3 Quantum Physics F2005 6 Momentum in square well One part of the momentum distribution looks like this The other looks like this = + peaked at p = ±h$ / L The sum see phi_p for square_well.mws Notes 3 Quantum Physics F2005 7 Momentum in square well Now to calculate expectation value of momentum, 2 $ L $ L sin p & p 2 2h sin + % % L 2 2 h + pdp p = ∫ "* p"dp = ∫ 4$ h $ L $ L &% &% & p p + 2 2h 2 2 h (We don't really want to calculate this awful thing, so let's think about symmetry. We note that " is symmetric about p = 0. This means that p" is an odd function, and therefore the integral is zero.) The situation would be much worse if I asked you to calculate p 2 because you'd really have to do the integral. Notes 3 Quantum Physics F2005 8 The Momentum Operator: a plausibility argument We want to be able to do something like this: % px = ∫ ! * ( x, t ) px ! ( x, t ) dx &% but we don't yet know how to write p ( x). Consider a pure momentum function, which is used to make up a wavepacket: 1 ! p ( x, t ) = ei ( px & Et ) / h 2$ h to extract p from this, we might take the derivative w.r.t. x. '! p ip = !p h 'x Notes 3 ' ⇒ & i h ! p ( x, t ) = p ! p 'x Quantum Physics F2005 9 The Momentum Operator • From a plausibility argument for a pure momentum state, we found a “momentum operator” ' pˆ x ( &ih 'x We generalize this idea and will apply this operator to any wavefunction to extract information on the momentum. ' * ˆ px = ∫ ! ( x, t ) px ! ( x, t ) dx = ∫ ! ( x, t ) &ih ! ( x, t )dx 'x &% &% % Notes 3 * % Quantum Physics F2005 10 Momentum in the square well ' px = ∫ ! ( x, t ) pˆ x ! ( x, t ) dx = ∫ ! * ( x, t ) &ih ! ( x, t )dx 'x &% &% &ih 2 % $x ' $ x = ∫ cos cos dx L &% L 'x L % * % &ih 2$ % &ih 2$ % $x $x = ∫ cos ∫ udu = 0 sin dx = 2 2 L &% L &% L L Notes 3 Quantum Physics F2005 11 Where are we? (and where are we going?) • We are in the middle of Chapter 5, learning the concept of the operator for a quantum observable. • We will review a few of the most frequently used operators and some obvious ramifications of this approach. • By the end of this class, we will formally introduce the Schrodinger equation and begin to address some well-known solutions. – stationary states and the time-independent Schrodinger equation. – the time dependent SE. Notes 3 Quantum Physics F2005 12 A more general momentum operator derivation • following Morrison prob 5.1. A reasonable starting point, p =m d x dt We will also use Schrodinger's equation ih ' 2 ! * V * '! ih ' 2 ! V '! * = + ! and =& & ! 2 2 ih ih 't 2m 'x 't 2m 'x So: '! * ' % * * '! p = m ∫ ! x!dx = m ∫ dx x! + ! x 't &% 't 't 2 ih ' 2 ! * V * V * ih ' ! & ! x! + ! x + ! dx = m ∫ & 2 2 ih ih 2m 'x 2m 'x Notes 3 ih * ' 2 ! ' 2 ! * x! dx = ∫! x 2 & 2 2 'x 'x F2005 Quantum Physics 13 A more general momentum operator derivation ih * ' 2 ! ' 2 ! * p = ∫! x 2 & x! dx 2 2 'x 'x * '! ih ' * '! '! * * x! + ! ! & 2 ! = ∫ ! x & dx 2 'x 'x 'x 'x * % % ih * '! '! * * '! x! + ! ! & i h ∫ ! dx = ! x & 2 'x 'x 'x &% &% % '! p = ∫ ! ( &i h )dx = ∫ ! * pˆ !dx 'x &% &% '! ˆp ( &ih 'x % Notes 3 * Quantum Physics F2005 14 Momentum Operator Whenever a wavefunction !(x,t) is used to describe a state of a system, it is correct to represent the momentum variable by means of the differential operator: ' pˆ ) &ih 'x Notes 3 Quantum Physics F2005 15 The Operator Postulate • In quantum mechanics, every observable is represented by an operator that is used to obtain physical information about the observable from the state functions. Notes 3 Quantum Physics F2005 16 Operator mathematics • You already know about operators, you’ve used them in Math, Theo Phys, Optics… They are essentially a set of instructions on what to do to a function (take the derivative, multiply it by…) • Some rules: – Operators must act on a function – Operators act on every function to their right unless their action is constrained by brackets – The product of operators implies successive operation (starting with the rightmost). – The sequence of operations does not commute. – addition or subtration of operators means to distribute the operations Notes 3 Quantum Physics F2005 17 How general operators differ from the specialized ones you learned in elementary school Non-commuting operators ' Let's look at operators Qˆ1 and Qˆ 2 . Take Qˆ1 = pˆ = &ih and Qˆ 2 = xˆ = x. 'x Is Qˆ Qˆ f ( x) = Qˆ Qˆ f ( x) ? 1 2 2 1 ' 'f Qˆ1Qˆ 2 f ( x) = &ih [ xf ] = &ih f + x 'x 'x 'f Qˆ 2Qˆ1 f ( x) = &ihx 'x A measure of whether operators commute is the difference due to order: (Qˆ Qˆ & Qˆ Qˆ ) f = ( xpˆˆ & pxˆ ˆ ) f = ihf 2 1 1 2 Exercise 5 – Check this expression explicitly for f=e2x. Notes 3 Quantum Physics F2005 18 Exercise 5 - solution ' and Qˆ 2 = xˆ = x and f ( x) = e2 x . Qˆ1 = pˆ = &ih 'x Is Qˆ Qˆ f ( x) = Qˆ Qˆ f ( x) ? 1 2 2 1 ' Qˆ1Qˆ 2 f ( x) = &ih xe 2 x = &ih e 2 x + 2 xe 2 x 'x ' Qˆ 2Qˆ1 f ( x) = &ihx e 2 x = &ih 2 xe2 x 'x A measure of whether operators commute is the difference due to order: (Qˆ Qˆ & Qˆ Qˆ ) e 2 1 Notes 3 1 2 2x ˆˆ & px ˆ ˆ ) e 2 x = &ih 2 xe 2 x & &ih e2 x + 2 xe2 x = ihe2 x = ( xp Quantum Physics F2005 19 Some other operators To find the expectation value of a general observable: Q. First construct an operator to extract Q, Qˆ . (For wavefunctions, this can usually be accomplished by rewriting the physical quantity in terms of x and p.) Then compute the expectation integral as before Q = ∫ ! *Qˆ ( x)!dx Notes 3 Quantum Physics F2005 20 Some other operators 2 p Classical kinetic energy T = 2m (When an operator is squared, that means to perform the operation twice) 1 2 1 1 ' ' ˆ ˆ ˆ ˆ T* = p* = pp* = &ih &ih 'x 'x 2m 2m 2m &h2 ' 2 = * 2 2m 'x Position: xˆ = x Constant: Cˆ = C Notes 3 Quantum Physics F2005 21 Some other operators Potential energy, V : Vˆ Usually, the potential energy, V , is a function of x and t. so we just contruct it from V(x,t). Example- a spring: V = kx 2 / 2 ⇒ Vˆ = kxˆ 2 / 2 If it has time dependence we need to include that too. The total energy operator in x: Hˆ = Tˆ + Vˆ is called the Hamiltonian. ' ˆ The energy operator in terms of t : E = ih 't Notes 3 Quantum Physics F2005 22 The Schrodinger Equation • The Schrodinger equation is a relationship between the space and time representations of the energy operators. • This equation allows you to calculate the time-development of the state functions. Hˆ * = Eˆ* 2 2 h ' '* & * + V ( x, t )* = ih 2 't 2m 'x Notes 3 Quantum Physics F2005 23