� � � � � � � � � � � � � � � � � 8.04 Quantum Physics Lecture XIII(13-1) �p�∗ = �pˆ��� ���h¯∂ ∗ = dxΨ∗ Ψ (13-2) i ∂x h¯∂ = dxΨΨ∗ (13-3) − i ∂x h¯∞ ∂∂ = − dx (ΨΨ∗) − Ψ∗ Ψ (13-4) i ∂x ∂x � −∞�� h¯∂ = dxΨ∗ Ψ (13-5) i ∂x = �p� , (13-6) where again we have used integration by parts and 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 corresponding stationary state, all eigenvalues of the momentum operator must be real. An operator whose eigenvaluse are real (or equivalently, whose expectation value for all admissible wavefunctions is real) is called a Hermitian operator. Physically measurable quantities are represented by Hermitian operators.Similarly, one can show that �E�∗ = �E� for any state, so all energy eigenvalue are real: The Hamiltonian operator Hˆis a Hermitian operator. Can we “derive” Newton’s F = ma from the SE? CM: F dV = ma = dp Let us calculate the expectation value of dp := −dxdt dt dp d dt = dt �p� (13-7) dh¯∂ = dxΨ∗(x,t) Ψ(x,t) (13-8) dt i∂x h¯∂Ψ∗ ∂Ψ ∂∂ = dx +Ψ∗ (13-9) i ∂t ∂x ∂x ∂t �� ��� h¯2 ∂2Ψ∗ ∂ψ ∂Ψ ∂h¯2 ∂2Ψ = dx −2m ∂x2 ∂x + V Ψ∗ ∂x − Ψ∗ ∂x −2m ∂x2 + Vψ (13-10) A Massachusetts Institute of Technology XIII-1 + ���� + � � � � � � � � � � � � 8.04 Quantum Physics Lecture XIIIThe integrand is h¯2 ∂2Ψ∗ ∂Ψ ∂Ψ� h¯2 ∂∂2Ψ A = −2m ∂x2 ∂x V Ψ∗ ∂x 2m ∂x Ψ∗ ∂x2 h¯2 ∂Ψ∗ ∂2Ψ ∂V Ψ − ���∂�Ψ� (13-11) − 2m ∂x ∂x2 − Ψ∗ ∂x Ψ∗V ∂x h¯2 ∂∂Ψ ∂Ψ∗ ∂Ψ ∂V =2m ∂x Ψ∗ ∂x − ∂x ∂x − Ψ∗ ∂x Ψ (13-12) Again the integral over the first term vanishes since Ψ 0 for x → ±∞, and we are →left with ��� �� �� dp ∂V dV dt = dxΨ∗(x,t) − ∂x (x) Ψ(x,t)= − dx (13-13) or d dV (13-14) dt �p� = − dx It follows from the SE that the expectation values obey the classical equations of motion. d mdt �x� = �p��� (13-15) d dV dt �p� = − dx (13-16) Average momentum changes due to average force ��� ��� dV ∂V 2− dx = − dxΨ∗ ∂x Ψ= dxF (x)|Ψ(x,t)|, (13-17) i.e. position-dependent force F (x) is weighted by probability density Ψ(x,t)2 for dV |d |finding the particle at position x at time t. Note, however, that dx =�d�x� V (�x�) Example 1. Double-peaked distribution. The probability to find the particle at the average position �x� is small, so the force there cannot be of much consequence for the particle’s motion. Example 2. Force varying quickly on wavepacket scale. Classical calculation − d V (�x�)d�x�would predict very large (and quickly varying force as �x� changes), actual QM force Massachusetts Institute of Technology XIII-2 � � � � � � 8.04 Quantum Physics Lecture XIIIFigure I: Double-peaked particle distribution with vanishing probability to find partiicl at average position. Figure II: Particle wavepacket large compared to spatial variation of the force. The time evolution of the wavepacket will depend on the value of the force averaged over the wavepacket, not just on the force at the average particle location. dV−experienced by particle is much smaller. However, if the force varies slowly dx compared to the size of a single wavepacket, then dV d − dx = �F (x)�≈ F (�x�)= −d �x� V (�x�) (13-18) This is the reason why we can treat particles in macroscopic potential usually as classical particles. d dV → always true (13-19) dt �p� = − dx Massachusetts Institute of Technology XIII-3 � � � � � � � � � � � 8.04 Quantum Physics Lecture XIIIdd Ehrenfest’s theorem for slowly varying potentials dt �p�≈− d �x� V (�x�) → (13-20) Eigenfunctions of the momentum operator What are the eigenfunctions up of the momentum operator, i.e. the eigenfunctions satisfying puˆp = pup, (13-21) where p is some (fixed) particular eigenvalue of ˆp. We know that the operator ˆp is ¯∂Hermitian, so all eigenvalues p are real. In position space, we have pˆ= hi ∂x and h Aeipx/¯¯∂ up(x)= pup(x), up(x)= h . The momentum eigenfunctions are (of course) i ∂x just the plane waves. Let us check the orthonormality condition for eigenstates: ∞ dxup ∗(x)up(x)= Ap� ∗Ap dxe−ip�x/h¯eipx/h¯(13-22) −∞ � = Ap� ∗Ap dxei(p−p�)x/¯h (13-23) =¯hAp ∗Ap dyei(p−p�)y (13-24) =¯hAp� ∗Apδ(p − p�)2π (13-25) =¯h2πAp� ∗Apδ(p − p�) (13-26) The momentum eigenfunction are orthogonal for p = p�, but we have a normalization problem for p = p�: The Dirac delta function diverfes, or equivalently, the integral ∞ dxup(x)2 = AP 2 dx eipx/¯h2 (13-27) ||||−∞ |� � |� =1 diverges. Before looking at possibilities to deal with normalization problem, let us calculate the expansion coefficients c(p) c(p)= dxAp ∗e−ipx/¯hψ(x)= Ap ∗ dxψ(x)e−ipx¯h (13-28) √2π¯hφ(p) Massachusetts Institute of Technology XIII-4 � � � � � � 8.04 Quantum Physics Lecture XIIIWe see that if we make the normalization choice Ap = √21 π¯h , then the expansion coefficients c(p) into momentum eigenstates are just given by the Fourier transform 1 up(x)= eipx/¯h ”normalized” momentum eigenstates (13-29) √2πh¯→ ∞ ψ(x)= dpφ(p)up(x) (13-30) �−∞ � � expansion into momentum eigenstates 1 = dpφ(p)eipx/¯h (13-31) √2πh¯Fourier transformation The expansion into momentum eigenstates and Fourier tranformation are one and the same. Since ψ(x) and φ(p) contain the same information about the particle, we can use either one to characterize the position and motion of the particle. A more fundamental motion is the state of the particle (a state is a vector in Hilbert space), the state can be expressed (written down) in various representations (like position representation ψ(x), momentum representation φ(p), energy representation cE ) associated with Hermitian operators (position ˆx, momentum ˆp, energy Hˆ). We call φ(p) the momentum representation of a particular state, and 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) dimensions, the Hilbert space is one-(three-) dimensioonal but for N particles in three dimensions the Hilbert space is 3N-dimensional. In general, it cannot be factored into a tensor product of N three-dimensional vector space Vsystem =�V1 ⊗ V2 ⊗···⊗ VN , or equivalently, the wavefunction for N particles does not factor into a product of wavefunctions for each particle, Ψ(r1, r2,..., r1,t)=Ψ1(r1)Ψ2(r2) ... ΨN (rN) (13-32) In this case, when the wavefunction for an N-particle system cannot be written as a product of wavefunctions for the individual particles, i.e. when the particles do not evolve independently, we speak of an entangled 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 associated with entanglement, and the system behaves quasi-classically. A quantum system that could preserve the correlations, and that could be manipulaate externally, would constitute a quantum computer. A quantum computer could solve certain computation problems (only a handful have been discovered so far) exponenntiall faster than a classical computer. Because of the enormous size of the Massachusetts Institute of Technology XIII-5 8.04 Quantum Physics Lecture XIIIHilbert space, certain quantum mechanical problems involving many-particle correlattion (e.g. high temperature superconductivity that involves correlated motion of many electrons) are very difficult to solve or simulate on a classical computer. Now back to a single particle in one dimension . . . Massachusetts Institute of Technology XIII-6