Understanding Eigenstates & Eigenvalues in Quantum Mechanics: Operators & Commutation, Study notes of Quantum Physics

Download Understanding Eigenstates & Eigenvalues in Quantum Mechanics: Operators & Commutation and more Study notes Quantum Physics in PDF only on Docsity! Page 1 Physics 486 Lecture 6 Physics 486, Spring ‘07 Lecture 6 x Properties of Eigenstates and Eigenvalues in Quantum Postulates of Quantum Mechanics Let’s review the “postulates” that tell us how to use the theory to predict the results of experiments: Operator: (2). Postulate 2: Every observable (measurable) quantity in a quantum mechanical system (e.g., energy, position, momentum, etc.) has an operator associated with it: 2 22 2 dx d m T −= ( )xV dx d m H +−= 2 22 2 xxop =position dx d i pop =momentum Observable quantity: kinetic energy total energy (1). Postulate 1: The state of any physical system is specified, at each time t, by a state function Ψ(t). This state function describes all the information we can obtain about the system. Quantum Mechanical Operators We found that operators associated with observable quantities, i.e., those that we can measure in quantum systems, are hermitian, which is a fancy way of saying that their expectation values are real (rather than complex) quantities: *AA = ( ) dxxxAAdxxAxA )()()()( *** ΨΨ==ΨΨ= ∫∫ ∞ ∞− ∞ ∞− If an expectation value is real: Quantum Mechanical Operators Postulate 3: Another important and general postulate of quantum mechanics is that “in each measurement of an observable quantity, represented by an operator A, the only possible results of the measurement are the eigenvalues a of A”, where Aψ=aψ (for a primer on eigenvalues and eigenfunctions, see slide after next). )()()()( 2 2 22 xExxV dx xd m ψψψ =+− KE term PE term Total E term The time-independent SEQ can be more simply written as a linear “operator”: ( ) ( )H x E xψ ψ= 2 2 2 ( )2 dH V x m dx = − +Where H is the “energy” operator – called the “Hamiltonian” – which is given by: eigenstate eigenvalue Important example: Page 2 Physics 486 Lecture 6 Quantum Mechanical Operators ( ) L xn L xn πψ sin2= 2 2 2 2 2 12 22 2n n h nE n E m mL π λ = = = Example: Energy eigenstates and eigenvalues of infinite square well potential (eigenstates) (energy eigenvalues) n n nH Eψ ψ= Primer on Operators II: Eigenfunctions and Eigenvalues Suppose that for a particular linear operator L, there is a function f(x) that has the property: Example: the function f(x) = Aecx, where A and c are constants, is an eigenfunction of the derivative operator D, with eigenvalue c: where c is a real or complex number. L[f(x)] = cf(x) In this special situation, f(x) is called the eigenfunction of operator L, and c is the associated eigenvalue. The specific set of eigenfunctions and eigenvalues depends on the operator L. D[Aecx] = c(Aecx) In other words, applying the operator L to f(x) simply gives the function f(x) back again, but multiplied by the constant c. Quantum Mechanical Operators ( ) /ikx ipxx Ae Aeψ = = Example: Eigenstates of the momentum operator Consider plane wave solutions: opp ψ λψ= ( ) ( ) ( )/ / /ipx ipx ipxop dp e e p ei dx= = So, plane waves are eigenstates of the momentum operator, ( ) /ikx ipxx Ae Aeψ = = With eigenvalues (i.e., possible measured values of momentum): p hkλ = = Note that this is a continuous spectrum of eigenstates, because there are no boundary conditions that put constraints on k Example: Eigenstates and eigenvalues of parity operator P ( ) ( )P x xψ ψ= − So, we’re looking for solutions of the following form: ( ) ( )P x xψ λψ= Quantum Mechanical Operators Combining (*) and (**) gives: The parity operator P converts x to –x in a function: (*) (**) ( ) ( )x xψ λψ− = Now, if we apply the parity operator to both sides, we get: ( ) ( ) ( ) ( ) ( )2P x x P x P x xψ ψ λψ λ ψ λ ψ− = = = =   So, we find that: ( ) ( )2x xψ λ ψ= 1λ = ± are the eigenvaluesof the parity operator Page 5 Physics 486 Lecture 6 General Properties of Eigenstates V=∞ ψ(x) 0 L V=∞n=1 n=2 x n=3 ( ) L xn L xn πψ sin2= Example: consider the eigenstates of the infinite square well potential ( ) ( ) ( ) ( ) dx L xnm L xnm L dx L xn L xm L dxxx LL nm     +− − == ∫∫∫ ππππψψ coscos1sinsin2 00 * ( ) ( ) ( ) ( ) L L xnm nmL xnm nm 0 sin1sin1       + + − − − = π π π π ( ) ( ) ( )( ) ( ) ( ) mnnm nm nm nm nmdxxx , * sinsin1 δππ π ψψ =      + + − − − =∫ = 0 when n ≠ m = 1 when n = m Applying orthogonality condition: General Properties of Eigenstates Property 3: It is a postulate* of QM that eigenstates of the time- independent SEQ form a complete set of states, in the sense that any other function f(x) can be expressed as a linear combination of the eigenstates ψn (i.e., they “provide a suitable basis set” or “span the vector space”): ( )xcxf nn n ψ∑ = = 1 )( For example, in the case of the eigenstates of the infinite square well potential: V=∞ ψ(x) 0 L V=∞n=1 n=2 x n=3 ( ) L xn L xn πψ sin2= The expansion in (*) above is easily recognized as the Fourier series: L xnc L xf n n πsin2)( 1 ∑ = = *The proof that any function can be expanded in terms of a Fourier series is called Dirichlet’s theorem, and is beyond the scope of this course. There is no general proof of this for an infinite-dimensional vector space - although it is assumed to be true in quantum – which is why it’s a postulate Superpositions of Eigenstates • Remember from our earlier discussion that eigenstates of the time-independent SEQ are complete, i.e., any other function f(x) can be expressed as a linear combination of the eigenstates, ψn: ( )∑ = = 1 )( n nn xcxf ψ i.e., a wavefunction Ψ(x,t) representing the state of a quantum particle can be constructed from an appropriate superposition of eigenstates, ϕn(x,t)! The superposition principle says that such superpositions are ALSO solutions of the time-dependent SEQ! ( )∑=Ψ n nn txctx ,),( ϕ Important note: This wavefunction IS a solution to the time-dependent SEQ, but IS NOT a solution to (i.e., an eigenstate of) the time- independent SEQ, because it is associated with a superposition of energy states, NOT a single well-defined energy, and hence does not satisfy the eigenstate relation: HΨ=EΨ. Superpositions of Eigenstates Recall, though, that if we measure some observable quantity of the quantum particle, e.g., its position or energy, then we expect the particle to behave like a particle, in the sense that it will be measured at a particular position x or energy E. This measurement process is often referred to as the “collapse” of the particle wavefunction Ψ(x,t) into one of its eigenstates ϕn(x,t)! ( )∑=Ψ n nn txctx ,),( ϕ What does the superposition above mean physically? That the “wavelike” character of a quantum particle allows for the possibility that the particle in a given potential can be simultaneously in a mixture of different eigenstates and energies! The probability that a measurement will produce the eigenvalue of a particular eigenstate ϕn(x,t) is related to the coefficient cn. Page 6 Physics 486 Lecture 6 Superpositions of Eigenstates ( )∑=Ψ n nn txctx ,),( ϕ From the previous discussion, we clearly need a prescription for finding the coefficient cn, assuming that we know the particle wavefunction Ψ(x,t) and the normalized eigenstates of the potential in which the particle is confined, ϕn(x,t). ( )∑=Ψ n nn xcx ϕ)(Let t=0, and start with , multiply both sides by ϕ*m(x,t), then integrate: ( ) ( ) ( ) ( ) ( )∫∑∫∑∫ ∞ ∞− ∞ ∞− ∞ ∞− ==Ψ dxxxcdxxxcdxxx nm n n n nmnm ϕϕϕϕϕ *** )( δn,m = 0 when n ≠ m= 1 when n = m Where we have used the fact that eigenstates of the SEQ are orthogonal. So: ( ) mmn n nm ccdxxx ==Ψ ∑∫ ∞ ∞− , * )( δϕ ( ) dxxxc mm )(* Ψ= ∫ ∞ ∞− ϕ Abstract Vector Space In the above relationship, it is worth noting the similarity with vector properties. A vector v can be expanded in terms of a linear combination of orthogonal unit vectors, ei: ( ) dxxxc nn )(* Ψ= ∫ ∞ ∞− ϕ ( ) dxxxc mm )(* Ψ= ∫ ∞ ∞− ϕ i i i ecv ˆ∑= Vectors In turns out that – because they are orthogonal, form a complete set of states, and have the other general properties we associate with vectors - we can treat orthogonal functions such as the eigenstates ψn(x) as comprising an “abstract” vector space: ( )( ) n n n x c xϕΨ =∑ where the projection ci of the vector v onto a particular unit vector ei is given by: vec ii ⋅= ˆ Eigenstates Note: the coefficient cncan be thought of as the “projection” of the wavefunction Ψ(x) onto the nth eigenstate ϕn(x)! where Abstract Vector Space: Hilbert Space In fact, the set of wavefunctions in which we’re interested in quantum mechanics is more restricted then the vector space of all functions, because we require that the wavefunctions be normalizable: *( ) ( ) 1x x dx ∞ −∞ Ψ Ψ =∫ The specific class of functions that satisfy the above property constitutes a Hilbert space. Consequently, in quantum mechanics, eigenstates of the time-independent SEQ constitute a Hilbert space.