Quantum Physics Notes-7 Operators, Observables, Understanding QM. Notes 6 Quantum Physics F2005 1

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1 Quantum Physics 2005 Notes-7 Operators, Observables, Understanding QM Notes 6 Quantum Physics F2005

2 A summary of this section This section of notes is a brief overview of the ideas in chapters 0-2 of Morrison. These chapters are the intellectual core of uantum mechanics (as opposed to uantum physics). You will address the material in chapters 0-2 with greater care in Intro to Quantum Mechanics. The purpose of these notes is only to introduce these ideas, especially for those students who will not take Intro to Quantum Mechanics. I will therefore skip lightly through these chapters, picking only the most important concepts. Notes 6 Quantum Physics F2005 2

3 Operators Dirac notation Dirac Notation - "bra"= "ket"= Q =! Qˆ! =! Qˆ! dx * allspace! Qˆ! =! Qˆ! dx * 2 2 allspace * 2 allspace!! =!! dx =expectation value =matrix element =overlap integral ) Take the complex conjugate of the function inside the 2) Act with the operator on the function to its right. 3) Integrate the integrand over all space. Notes 6 Quantum Physics F2005 3

4 Hermitian Operators Qˆ is Hermitian if for any two physically admissible state functions! Qˆ! = Qˆ!! 2 2 ( * ) * Qˆ!! ˆ 2 " Q!! 2dx If an operator is Hermitian, the expectation value of that operator will be real. All operators for physical uantities are Hermitian. Notes 6 Quantum Physics F2005 4

5 Is p Hermitian? We will check explicitly whether % pˆ = $ ih % x Start by integrating! pˆ! 2! pˆ! = pˆ!!. by parts. 2 2 # # * % * # % *! ˆ p! 2 =! $ ih! 2 dx i 2 2 dx x = $ h!! $!! $# $# % $# % x # % # * % = ih! 2! dx =! 2 $ i! $# % x h % x $# Therefore p ˆ is Hermitian. * dx = pˆ!! 2 Notes 6 Quantum Physics F2005 5

6 Exercise Show by writing out the real space integral representation that the operator x ˆ is Hermitian. Notes 6 Quantum Physics F2005 6

7 Operators and Eigenvalues In uantum physics, the form of the eigenvalue euation for an observable Aˆ with eigenstate & and eigenvalue a is: Aˆ & = a&. a The collection of all eigenvalues is called the spectrum of this operator. a The most familiar eigenstate is the stationary state. & $ iet / h which satisfies: ˆ! e H E E = E! =! In an eigenstate of an observable, the uncertainty in that observable is zero. Notes 6 Quantum Physics F2005 7

8 The eigenstate of one observable might not be an eigenstate of another observable (but it can happen ). Example- An energy eigenstate of the suare well is: 2 ( x E h( ( x, t) Acos e with ' L h 2mL $ i' t! = = = % Is this an energy eigenstate of momentum pˆ = $ ih? % x % ( x $ i' t ( ( x $ i' t pˆ! = $ ih Acos e = iha sin e ) p! NO! % x L L L Another example: An energy eigenstate of the free wave is:! = % i( kx-' t) pˆ! = $ ih Ae = hk! YES> % x ( -' t) Notes 6 Quantum Physics F Ae i kx.

9 Hermiticity: Real eigenvalues and orthonormal eigenfunctions The eigenvalues of a Hermitian operator are real. The eigenvalues of a Hermitian operator are the only values that we can observe in a measurement of that observable. AND The eigenfunctions of a Hermitian operator are orthogonal, constitute a complete set, and satisfy closure. (p 465, Morrison). Notes 6 Quantum Physics F2005 9

10 Proof of orthogonality Take two different eigenstates of the same observable: ) Qˆ & = & and 2) Qˆ & = '& ' ' We want to demonstrate that & Qˆ & = & & = ' ' Notes 6 Quantum Physics F ' = 0. Multiply both sides of ) by complex conjugate of & : Use Hermiticity of Q ˆ to simplfy the LHS: & Qˆ & = Qˆ & & = '& & = '* & & = ' & & ' ' ' ' ' So we have so & ' & ( ) = 0 ' & - ' & & = 0, but we assumed ) ' & '

11 Hermiticity: Completeness Completeness of a set of eigenfunctions means that any well-behaved function of the variables on which the eigenfunctions depend can be expanded upon the set. ) f = c & We can choose to represent any function in a given range of variables by an expansion in a set of eigenfunctions of any operator. It's easy to find the coefficients ' ' ' ' Notes 6 Quantum Physics F2005 c. Multiply ) by & f = & c & = c & c & = c The coefficient c is the "projection" of f onto the eigenfunction & & '

12 Summarizing Eigenfunction Expansions Notes 6 Quantum Physics F2005 2

13 The meaning and use of expansions The collection of coefficients in the expansion of a state function in any complete set is merely an alternate way to represent the state function. These coefficients and the eigenfunctions contain the same information as the state function. Expressing a state function in terms of eigenfunctions can make apparent some properties of the state function. Expressing the state function in terms of eigenfunctions can allow us to get information about other variables. (Expressing the wavefunction (position variable) in terms of momentum eigenfunctions allows us to determine the momentum properties of the state function.) Notes 6 Quantum Physics F2005 3

14 Using expansions to calculate expectation values If we have a wavefunction that is not an eigenfunction of a given operator, the expectation value may be hard to calculate in the old way. Expanding the wavefunction in terms of the eigenfunctions of that operator may make things easier. Notes 6 Quantum Physics F2005 4

15 Calculation of the mean Q ( t) =! * Qˆ! dx [ & ] = c t x Qˆ c t x dx * n' ( ) n' ( ) n ( )& n( ) * [ ( )& ( )] [ ( ) & ( )] = c t x c t x dx n' n' n n n * 2 cn' cn & n' & n cn n n' n n = = If we know the projection coefficients, we know the expectation value. Notes 6 Quantum Physics F2005 5

16 Example Calculate the expected value of the energy for the following non-stationary state of the SHO. $ i' t $ i' 2t!( x, t) = & e + & e $ i' 7 t $ i' 2t c ( t) = e ; c2( t) = e (all others=0) # E ( t) = cn En = h' 0 + h' 0 = h' 0 n= Note that any individual measurement of E will yield an eigenenergy. Notes 6 Quantum Physics F2005 6

17 The Commutator A simultaneous eigenstate is one whose state function is an eigenstate of two operators: Qˆ & = & and Rˆ & = r&, r, r, r, r Because they are eigenstates: Q = * Q = 0 and R = r * R = 0 Qˆ, Rˆ = QR ˆ ˆ - RQ ˆ ˆ = 0 ) Operators that commute define a complete set of simultaneous eigenfunctions. 2) Two operators that share a complete set of eigenfunctions commute. Notes 6 Quantum Physics F2005 7

18 Commutators and uncertainty principles If operators R ˆ and Q ˆ commute, then they share simultaneous eigenfunctions. Measurement of each observable for an eigenfunction yields a precise value. The only uncertainty relation that can apply is: * Q* R, 0. A Generalized Uncertainty Principle: * Q* R, 2 i Qˆ, Rˆ (For momentum and position: [ xˆ pˆ ], = ih) ˆ If Qˆ, Rˆ ) 0 then there exist no eigenstates of either observable. (Think about + ( x) and e - i( kx-' t ).) Notes 6 Quantum Physics F2005 8

19 Commutators of x and p Notes 6 Quantum Physics F2005 9

20 An aside Charles Hermite (December 24, January 4, 90) was a French mathematician who did research on number theory, uadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite polynomials, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor. He was the first to prove that e, the base of natural logarithms, is a transcendental number. His methods were later used by Ferdinand von Lindemann for the proof of his celebrated theorem that π is transcendental. Upon Weierstrass' discovery in 86 of continuous curves that are nowhere differentiable - they possess no tangent at any point - Hermite famously remarked: I turn aside with a shudder of horror from this lamentable plague of functions which have no derivatives. Notes 6 Quantum Physics F