Can we derive Newton s F = ma from the SE?

Transcription

1 8.04 Quantum Physics Lecture XIII p = pˆ (13-1) ( ( ) ) = xψ Ψ (13-) ( ) = xψ Ψ (13-3) [ ] = x (ΨΨ ) Ψ Ψ (13-4) ( ) = xψ Ψ (13-5) = p, (13-6) where again we have use integration by parts an the fact that Ψ vanishes at ±. Consequently, p = p, i.e. all expectation values of ˆp are real. Since an eigenvalue is the expectation value for the corresponing stationary state, all eigenvalues of the momentum operator must be real. An operator whose eigenvaluse are real (or equivalently, whose expectation value for all amissible wavefunctions is real) is calle a Hermitian operator. Physically measurable quantities are represente by Hermitian operators. Similarly, one can show that E = E for any state, so all energy eigenvalue are real: The Hamiltonian operator Ĥ is a Hermitian operator. Can we erive Newton s F = ma from the SE? V CM: F = = ma = p Let us calculate the expectation value of p x t t : p = p (13-7) t t ( ) = xψ (x, t) Ψ(x, t) (13-8) t ( ) Ψ Ψ = x + Ψ (13-9) i t t [ ( )] h Ψ ψ Ψ h Ψ = x m + V Ψ Ψ + V ψ (13-10) m }{{} A XIII-1

2 8.04 Quantum Physics Lecture XIII The integran is ( ) h Ψ Ψ + Ψ h Ψ A = V Ψ + Ψ m m ( ) h Ψ Ψ V Ψ Ψ (13-11) m Ψ Ψ V [ ] ( ) h Ψ Ψ Ψ V = Ψ m Ψ Ψ (13-1) Again the integral over the first term vanishes since Ψ 0 for x ±, an we are left with ( ) p V V = xψ (x, t) (x) Ψ(x, t) = (13-13) t x or V t p = x (13-14) It follows from the SE that the expectation values obey the classical equations of motion. m x = p t V t p = x (13-15) (13-16) Average momentum changes ue to average force ( ) V V = xψ Ψ = xf (x) Ψ(x, t), (13-17) x i.e. position-epenent force F (x) is weighte by probability ensity Ψ(x, t) for V fining the particle at position x at time t. Note, however, that = V ( x ) x x Example 1. Double-peake istribution. The probability to fin the particle at the average position x is small, so the force there cannot be of much consequence for the particle s motion. Example. Force varying quickly on wavepacket scale. Classical calculation V ( x ) x woul preict very large (an quickly varying force as x changes), actual QM force XIII-

3 8.04 Quantum Physics Lecture XIII Figure I: Double-peake particle istribution with vanishing probability to fin particle at average position. Figure II: Particle wavepacket large compare to spatial variation of the force. The time evolution of the wavepacket will epen on the value of the force average over the wavepacket, not just on the force at the average particle location. V experience by particle is much smaller. However, if the force varies slowly x compare to the size of a single wavepacket, then V = F (x) F ( x ) = V ( x ) (13-18) x x This is the reason why we can treat particles in macroscopic potential usually as classical particles. V always true (13-19) t p = x XIII-3

4 8.04 Quantum Physics Lecture XIII t p x V ( x ) Ehrenfest s theorem for slowly varying potentials (13-0) Eigenfunctions of the momentum operator What are the eigenfunctions u p of the momentum operator, i.e. the eigenfunctions satisfying pu ˆ p = pu p, (13-1) where p is some (fixe) particular eigenvalue of ˆp. We know that the operator ˆp is Hermitian, so all eigenvalues p are real. In position space, we have pˆ = h an h u Ae ipx/ h p (x) = pu p (x), u p (x) =. The momentum eigenfunctions are (of course) just the plane waves. Let us check the orthonormality conition for eigenstates: xu p (x)u p (x) = A p A p xe ip x/h e ipx/h (13-) = A p A p xe i(p p )x/ h (13-3) = ha p A p ye i(p p )y (13-4) x u p (x) = A P = ha p A p δ(p p )π (13-5) = hπa p A p δ(p p ) (13-6) The momentum eigenfunction are orthogonal for p = p, but we have a normalization problem for p = p : The Dirac elta function iverfes, or equivalently, the integral x e ipx/ h (13-7) }{{ } =1 iverges. Before looking at possibilities to eal with normalization problem, let us calculate the expansion coefficients c(p) c(p) = xa p e ipx/ h ψ(x) = A p xψ(x)e ipx h (13-8) }{{} π hφ(p) XIII-4

5 8.04 Quantum Physics Lecture XIII We see that if we make the normalization choice A p = 1, then the expansion π h coefficients c(p) into momentum eigenstates are just given by the Fourier transform 1 u p (x) = e ipx/ h normalize momentum eigenstates (13-9) πh ψ(x) = pφ(p)u p (x) (13-30) } {{} expansion into momentum eigenstates 1 = pφ(p)e ipx/ h (13-31) } πh {{} Fourier transformation The expansion into momentum eigenstates an Fourier tranformation are one an the same. Since ψ(x) an φ(p) contain the same information about the particle, we can use either one to characterize the position an motion of the particle. A more funamental motion is the state of the particle (a state is a vector in Hilbert space), the state can be expresse (written own) in various representations (like position representation ψ(x), momentum representation φ(p), energy representation c E ) associate with Hermitian operators (position ˆx, momentum ˆp, energy Ĥ). We call φ(p) the momentum representation of a particular state, an interpret it as the wavefunction in momentum space. The SE governs the time evolution of the wavefunction, or equivalently, the time evolution of the state of the particle in Hilbert space. For one particle in one (three) imensions, the Hilbert space is one- (three-) imensional, but for N particles in three imensions the Hilbert space is 3N-imensional. In general, it cannot be factore into a tensor prouct of N three-imensional vector space V system = V 1 V V N, or equivalently, the wavefunction for N particles oes not factor into a prouct of wavefunctions for each particle, Ψ(r 1, r,..., r 1, t) = Ψ 1 (r 1 )Ψ (r )... Ψ N (r N ) (13-3) In this case, when the wavefunction for an N-particle system cannot be written as a prouct of wavefunctions for the iniviual particles, i.e. when the particles o not evolve inepenently, we speak of an entangle state. Because of this possibility a quantum system of N particles is vastly (exponentially in N) richer than an classical system of N particles. However, in most cases we lose track of the particle-particle correlations associate with entanglement, an the system behaves quasi-classically. A quantum system that coul preserve the correlations, an that coul be manipulate externally, woul constitute a quantum computer. A quantum computer coul solve certain computation problems (only a hanful have been iscovere so far) exponentially faster than a classical computer. Because of the enormous size of the XIII-5

6 8.04 Quantum Physics Lecture XIII Hilbert space, certain quantum mechanical problems involving many-particle correlations (e.g. high temperature superconuctivity that involves correlate motion of many electrons) are very ifficult to solve or simulate on a classical computer. Now back to a single particle in one imension... XIII-6