DOCSLIB.ORG

Unitary Time Evolution

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Unitary Time Evolution Unitary time evolution Time evolution of quantum systems is always given by Unitary Transformations. If the state of a quantum system is ψ , then at a later time | i ψ Uˆ ψ . | i→ | i Exactly what this operator Uˆ is will depend on the particular system and the interactions that it undergoes. It does not, however, depend on the state ψ . This | i means that time evolution of quantum systems is linear. Because of this linearity, if a system is in state ψ or φ or any linear combination, the time evolution is| giveni | byi the same operator: (α ψ + β φ ) Uˆ(α ψ + β φ ) = αUˆ ψ + βUˆ φ . | i | i → | i | i | i | i – p. 1/25 The Schrödinger equation As we have seen, these unitary operators arise from the Schrodinger¨ equation d ψ /dt = iHˆ (t) ψ /~, | i − | i where Hˆ (t) = Hˆ †(t) is the Hamiltonian of the system. Because this is a linear equation, the time evolution must be a linear transformation. We can prove that this must be a unitary transformation very simply. – p. 2/25 Suppose ψ(t) = Uˆ(t) ψ(0) for some matrix Uˆ(t) (which we don’t yet assume| i to be| unitary).i Plugging this into the Schrödinger equation gives us: dUˆ(t) dUˆ †(t) = iHˆ (t)Uˆ(t)/~, = iUˆ †(t)Hˆ (t)/~. dt − dt At t = 0, Uˆ(0) = Iˆ, so Uˆ †(0)Uˆ(0) = Iˆ. We see that d 1 Uˆ †(t)Uˆ(t) = Uˆ †(t) iHˆ (t) iHˆ (t) Uˆ(t) = 0. dt ~ − So Uˆ †(t)Uˆ(t) = Iˆ at all times t, and Uˆ(t) must always be unitary. – p. 3/25 For time-independent Hamiltonians we can easily write down the solution to the Schrödinger equation. Using the spectral theorem, we choose a basis k of eigenvectors of {| i} Hˆ with eigenvalues Ek, Hˆ k = Ek k . We then write ψ(t) in terms of this basis: | i | i | i 2 Hˆ = E k k , ψ(t) = α (t) k , α (t) = 1. k| ih | | i k | i | k | Xk Xk Xk Knowing ψ(t) = exp( iHt/ˆ ~) ψ(0) for any t means know- | i − | i ing the amplitudes αk(t). From the Schrödinger equation, dα k = iE α /~ = α (t) = exp( iE t/~)α (0). dt − k k ⇒ k − k k Each energy eigenstates undergoes a steady phase rotation. – p. 4/25 Bloch sphere rotation Any 2 2 Hermitian operator can be written Hˆ = aIˆ+ bXˆ+ × cYˆ + dZˆ with real a,b,c,d. Spin-1/2 unitaries take the form Uˆ(t) = exp (it/~)(aIˆ+ bXˆ + cYˆ + dZˆ) − We now need to use a very useful and important fact. For general operators Aˆ and Bˆ, usually exp(Aˆ)exp(Bˆ) = exp(Aˆ + Bˆ). 6 The one exception to this is when [A,ˆ Bˆ] = 0. In this case only, exp(Aˆ)exp(Bˆ) = exp(Aˆ + Bˆ). – p. 5/25 Since the identity commutes with everything, it exp aIˆ+ bXˆ + cYˆ + d − ~ iat it = exp Iˆ exp bXˆ + cYˆ + dZˆ − ~ − ~ ~ = e−iat/ exp (it/~)(bXˆ + cYˆ + dZˆ) . − Since an overall phase is meaningless, we can always set a = 0. (This is not just true for spin-1/2; one can add or subtract a term aIˆ to any Hamiltonian.) – p. 6/25 The most general spin-1/2 Hamiltonian is therefore Hˆ = bXˆ + cYˆ + dZˆ = E0~n ~σˆ · where 2 2 2 E0 = b + c + d , ~n =(nx,ny,nzp)=(b/E0,c/E0,d/E0), 2 2 2 ˆ ˆ ˆ ˆ with nx + ny + nz = 1 and ~σ =(X, Y, Z). The unitary is exp( iHt/ˆ ~) = cos(E0t/~)Iˆ i sin(E0t/~)~n ~σ.ˆ − − · In the Bloch sphere picture this corresponds to a rotation around the axis ~n at a rate E0/~. This is the most general unitary transformation possible for spin-1/2. – p. 7/25 Controlling unitaries We generalize from steady rotation by assuming we can turn the Hamiltonian on and off. By turning a Hamiltonian on for a particular length of time, we can “rotate” the state by a particular angle. For a spin-1/2, this means we can perform unitary transformations of the form Uˆ(θ) = cos(θ/2)Iˆ i sin(θ/2)~n ~σ.ˆ − · We can do the same with more complicated systems. For a D-dimensional system with a Hamiltonian Hˆ having eigenvalues E and eigenvectors k , we can do the unitary k | i Uˆ(τ) = exp( iE τ/~) k k − k | ih | Xk for any τ. – p. 8/25 Building up unitaries Unfortunately, we cannot always choose the exact values of the eigenvalues Ek or the eigenvectors k . These are generally given to us by nature. But we| sometimesi can increase the range of our options by combining several different unitaries in a row. The important thing to remember is that any product of unitary operators is also unitary: Uˆ †Uˆ = Vˆ †Vˆ = Iˆ = (UˆVˆ )†(UˆVˆ ) = Vˆ †Uˆ †UˆVˆ = I.ˆ ⇒ Suppose there are two different Hamiltonians we can turn on: Hˆ1 and Hˆ2. Then we can perform the unitaries Uˆ1(τ) = exp( iHˆ1τ/~), Uˆ2(τ) = exp( iHˆ2τ/~). − − But we can do much more than these! – p. 9/25 We can also do the unitaries Uˆ2(τ2)Uˆ1(τ1), and Uˆ2(τ3)Uˆ1(τ2)Uˆ2(τ1), and Uˆ2(τ )Uˆ1(τ −1) Uˆ2(τ2)Uˆ1(τ1). n n · · · Let’s see how this works for the spin-1/2. Suppose we can turn on Hamiltonians Hˆ1 = ExX,ˆ Hˆ2 = EyY.ˆ These produce unitaries Uˆ1(θ) and Uˆ2(θ) which correspond, in the Bloch sphere representation, to rotations by θ about the X and Y axes, respectively. – p. 10/25 There is a theorem in geometry that a rotation by any angle θ around any axis ~n can be done by doing three rotations in a row around the X and Y axes: R~n(θ) = RX(φ3)RY (φ2)RX(φ1) for some φ1,φ2,φ3. Since every 2 2 unitary is equivalent to a Bloch sphere rotation about× some axis ~n, any 2 2 unitary equals × UˆX (τ3)UˆY (τ2)UˆX(τ1) for some τ1,τ2,τ3 (up to an overall phase). When we talk about “turning on” and “turning off” Hamiltonians, what do we really mean? – p. 11/25 Here is a picture to give some idea: For many experimental systems, unitaries are effected by turning precisely-tuned lasers on and off for precise lengths of time. – p. 12/25 Tensor products of unitaries We have seen that Hilbert spaces of composite systems are represented by tensor products of the Hilbert spaces of the component systems: 12 = 1 2. H H ⊗ H Therefore, unitaries on the joint system also act on this larger Hilbert space. Suppose that the Hamiltonian of a system is a sum of terms which act only on the individual subsystems: Hˆ = Hˆ1 Iˆ+ Iˆ Hˆ2. ⊗ ⊗ The two terms represent the Hamiltonian of the first and second subsystems, respectively. In this Hamiltonian, the two subsystems are isolated, and do not interact. – p. 13/25 Hˆ1 Iˆ and Iˆ Hˆ2 commute, so ⊗ ⊗ Uˆ(t) = exp( iHt/ˆ ~) − = exp( i(Hˆ1 Iˆ+ Iˆ Hˆ2)t/~) − ⊗ ⊗ = exp( iHˆ1 It/ˆ ~)exp( iIˆ Hˆ2t/~) − ⊗ − ⊗ = exp( iHˆ1t/~) exp( iHˆ2t/~) − ⊗ − Uˆ1(t) Uˆ2(t). ≡ ⊗ It is a tensor product of unitaries. What if the Hamiltonian is not a sum of terms which act on the individual subsystems, but includes terms which act on both subsystems? – p. 14/25 Interactions and entanglement If the Hamiltonian has the form Hˆ = Hˆ1 Iˆ+ Iˆ Hˆ2 + Hˆint, ⊗ ⊗ the unitary Uˆ(t) = exp( iHt/ˆ ~) will not be a tensor product, in general. In− this case, we say that the two subsystems interact. If a tensor-product unitary Uˆ1 Uˆ2 acts on a product state ψ φ , then it will remain⊗ a product state. If a | i ⊗ | i general Uˆ acts on it, generally the state becomes entangled. Since most states are entangled, to produce them from initial product states requires unitaries which are not products. We must have interactions between the subsystems to produce general unitary transformations. – p. 15/25 Let’s take an example for two spin-1/2s: Hˆint = EintZˆ Zˆ. This yields unitary transformations of the form ⊗ Uˆ(θ) = cos(θ/2)Iˆ i sin(θ/2)Zˆ Z.ˆ − ⊗ Suppose we have an initial product state Ψ = ψ φ | i | i ⊗ | i = α1β1 + α1β2 + α2β1 + α2β2 . | ↑↑i | ↑↓i | ↓↑i | ↓↓i When we transform it by Uˆ(θ) it becomes −iθ/2 iθ/2 Uˆ(θ) Ψ = e α1β1 + e α1β2 | i | ↑↑i | ↑↓i iθ/2 −iθ/2 +e α2β1 + e α2β2 , | ↓↑i | ↓↓i which is no longer a product state for θ = mπ/2. The interaction has produced entanglement.6 – p. 16/25 The no-cloning theorem The restriction of time evolution to unitary operators means that certain kinds of evolution are impossible. One impossible task is quantum cloning. Suppose we have a system in an unknown state ψ , | i and we wish to copy it, i.e., to transform a second system starting in some standard state 0 into the same state | i ψ . Is there a unitary Uˆ such that, for any state ψ , | i | i Uˆ ( ψ 0 ) = ψ ψ ? | i ⊗ | i | i ⊗ | i If this is true, then also for φ = ψ | i 6 | i Uˆ ( φ 0 ) = φ φ . | i ⊗ | i | i ⊗ | i – p. 17/25 Consider now a superposition state χ = α ψ + β φ . By linearity, | i | i | i Uˆ( χ 0 ) = Uˆ(α ψ + β φ ) 0 | i ⊗ | i | i | i ⊗ | i = α ψ ψ + β φ φ | i ⊗ | i | i ⊗ | i = (α ψ + β φ ) (α ψ + β φ ), 6 | i | i ⊗ | i | i so Uˆ( χ 0 ) = χ χ , which is a contradiction.
Load more

Recommended publications
  • An Introduction to Quantum Field Theory
    AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
    [Show full text]
  • On the Time Evolution of Wavefunctions in Quantum Mechanics C
    On the Time Evolution of Wavefunctions in Quantum Mechanics C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology October 1999 1 Introduction The purpose of these notes is to help you appreciate the connection between eigenfunctions of the Hamiltonian and classical normal modes, and to help you understand the time propagator. 2 The Classical Coupled Mass Problem Here we will review the results of the coupled mass problem, Example 1.8.6 from Shankar. This is an example from classical physics which nevertheless demonstrates some of the essential features of coupled degrees of freedom in quantum mechanical problems and a general approach for removing such coupling. The problem involves two objects of equal mass, connected to two different walls andalsotoeachotherbysprings.UsingF = ma and Hooke’s Law( F = −kx) for the springs, and denoting the displacements of the two masses as x1 and x2, it is straightforward to deduce equations for the acceleration (second derivative in time,x ¨1 andx ¨2): k k x −2 x x ¨1 = m 1 + m 2 (1) k k x x − 2 x . ¨2 = m 1 m 2 (2) The goal of the problem is to solve these second-order differential equations to obtain the functions x1(t)andx2(t) describing the motion of the two masses at any given time. Since they are second-order differential equations, we need two initial conditions for each variable, i.e., x1(0), x˙ 1(0),x2(0), andx ˙ 2(0). Our two differential equations are clearly coupled,since¨x1 depends not only on x1, but also on x2 (and likewise forx ¨2).
    [Show full text]
  • An S-Matrix for Massless Particles Arxiv:1911.06821V2 [Hep-Th]
    An S-Matrix for Massless Particles Holmfridur Hannesdottir and Matthew D. Schwartz Department of Physics, Harvard University, Cambridge, MA 02138, USA Abstract The traditional S-matrix does not exist for theories with massless particles, such as quantum electrodynamics. The difficulty in isolating asymptotic states manifests itself as infrared divergences at each order in perturbation theory. Building on insights from the literature on coherent states and factorization, we construct an S-matrix that is free of singularities order-by-order in perturbation theory. Factorization guarantees that the asymptotic evolution in gauge theories is universal, i.e. independent of the hard process. Although the hard S-matrix element is computed between well-defined few particle Fock states, dressed/coherent states can be seen to form as intermediate states in the calculation of hard S-matrix elements. We present a framework for the perturbative calculation of hard S-matrix elements combining Lorentz-covariant Feyn- man rules for the dressed-state scattering with time-ordered perturbation theory for the asymptotic evolution. With hard cutoffs on the asymptotic Hamiltonian, the cancella- tion of divergences can be seen explicitly. In dimensional regularization, where the hard cutoffs are replaced by a renormalization scale, the contribution from the asymptotic evolution produces scaleless integrals that vanish. A number of illustrative examples are given in QED, QCD, and N = 4 super Yang-Mills theory. arXiv:1911.06821v2 [hep-th] 24 Aug 2020 Contents 1 Introduction1 2 The hard S-matrix6 2.1 SH and dressed states . .9 2.2 Computing observables using SH ........................... 11 2.3 Soft Wilson lines . 14 3 Computing the hard S-matrix 17 3.1 Asymptotic region Feynman rules .
    [Show full text]
  • Advanced Quantum Mechanics
    Advanced Quantum Mechanics Rajdeep Sensarma [email protected] Quantum Dynamics Lecture #9 Schrodinger and Heisenberg Picture Time Independent Hamiltonian Schrodinger Picture: A time evolving state in the Hilbert space with time independent operators iHtˆ i@t (t) = Hˆ (t) (t) = e− (0) | i | i | i | i i✏nt Eigenbasis of Hamiltonian: Hˆ n = ✏ n (t) = cn(t) n c (t)=c (0)e− n | i | i n n | i | i n X iHtˆ iHtˆ Operators and Expectation: A(t)= (t) Aˆ (t) = (0) e Aeˆ − (0) = (0) Aˆ(t) (0) h | | i h | | i h | | i Heisenberg Picture: A static initial state and time dependent operators iHtˆ iHtˆ Aˆ(t)=e Aeˆ − Schrodinger Picture Heisenberg Picture (0) (t) (0) (0) | i!| i | i!| i Equivalent description iHtˆ iHtˆ Aˆ Aˆ Aˆ Aˆ(t)=e Aeˆ − ! ! of a quantum system i@ (t) = Hˆ (t) i@ Aˆ(t)=[Aˆ(t), Hˆ ] t| i | i t Time Evolution and Propagator iHtˆ (t) = e− (0) Time Evolution Operator | i | i (x, t)= x (t) = x Uˆ(t) (0) = dx0 x Uˆ(t) x0 x0 (0) h | i h | | i h | | ih | i Z Propagator: Example : Free Particle The propagator satisfies and hence is often called the Green’s function Retarded and Advanced Propagator The following propagators are useful in different contexts Retarded or Causal Propagator: This propagates states forward in time for t > t’ Advanced or Anti-Causal Propagator: This propagates states backward in time for t < t’ Both the retarded and the advanced propagator satisfies the same diff. eqn., but with different boundary conditions Propagators in Frequency Space Energy Eigenbasis |n> The integral is ill defined due to oscillatory nature
    [Show full text]
  • The Liouville Equation in Atmospheric Predictability
    The Liouville Equation in Atmospheric Predictability Martin Ehrendorfer Institut fur¨ Meteorologie und Geophysik, Universitat¨ Innsbruck Innrain 52, A–6020 Innsbruck, Austria [email protected] 1 Introduction and Motivation It is widely recognized that weather forecasts made with dynamical models of the atmosphere are in- herently uncertain. Such uncertainty of forecasts produced with numerical weather prediction (NWP) models arises primarily from two sources: namely, from imperfect knowledge of the initial model condi- tions and from imperfections in the model formulation itself. The recognition of the potential importance of accurate initial model conditions and an accurate model formulation dates back to times even prior to operational NWP (Bjerknes 1904; Thompson 1957). In the context of NWP, the importance of these error sources in degrading the quality of forecasts was demonstrated to arise because errors introduced in atmospheric models, are, in general, growing (Lorenz 1982; Lorenz 1963; Lorenz 1993), which at the same time implies that the predictability of the atmosphere is subject to limitations (see, Errico et al. 2002). An example of the amplification of small errors in the initial conditions, or, equivalently, the di- vergence of initially nearby trajectories is given in Fig. 1, for the system discussed by Lorenz (1984). The uncertainty introduced into forecasts through uncertain initial model conditions, and uncertainties in model formulations, has been the subject of numerous studies carried out in parallel to the continuous development of NWP models (e.g., Leith 1974; Epstein 1969; Palmer 2000). In addition to studying the intrinsic predictability of the atmosphere (e.g., Lorenz 1969a; Lorenz 1969b; Thompson 1985a; Thompson 1985b), efforts have been directed at the quantification or predic- tion of forecast uncertainty that arises due to the sources of uncertainty mentioned above (see the review papers by Ehrendorfer 1997 and Palmer 2000, and Ehrendorfer 1999).
    [Show full text]
  • Chapter 6. Time Evolution in Quantum Mechanics
    6. Time Evolution in Quantum Mechanics 6.1 Time-dependent Schrodinger¨ equation 6.1.1 Solutions to the Schr¨odinger equation 6.1.2 Unitary Evolution 6.2 Evolution of wave-packets 6.3 Evolution of operators and expectation values 6.3.1 Heisenberg Equation 6.3.2 Ehrenfest’s theorem 6.4 Fermi’s Golden Rule Until now we used quantum mechanics to predict properties of atoms and nuclei. Since we were interested mostly in the equilibrium states of nuclei and in their energies, we only needed to look at a time-independent description of quantum-mechanical systems. To describe dynamical processes, such as radiation decays, scattering and nuclear reactions, we need to study how quantum mechanical systems evolve in time. 6.1 Time-dependent Schro¨dinger equation When we first introduced quantum mechanics, we saw that the fourth postulate of QM states that: The evolution of a closed system is unitary (reversible). The evolution is given by the time-dependent Schrodinger¨ equation ∂ ψ iI | ) = ψ ∂t H| ) where is the Hamiltonian of the system (the energy operator) and I is the reduced Planck constant (I = h/H2π with h the Planck constant, allowing conversion from energy to frequency units). We will focus mainly on the Schr¨odinger equation to describe the evolution of a quantum-mechanical system. The statement that the evolution of a closed quantum system is unitary is however more general. It means that the state of a system at a later time t is given by ψ(t) = U(t) ψ(0) , where U(t) is a unitary operator.
    [Show full text]
  • 2.-Time-Evolution-Operator-11-28
    2. TIME-EVOLUTION OPERATOR Dynamical processes in quantum mechanics are described by a Hamiltonian that depends on time. Naturally the question arises how do we deal with a time-dependent Hamiltonian? In principle, the time-dependent Schrödinger equation can be directly integrated choosing a basis set that spans the space of interest. Using a potential energy surface, one can propagate the system forward in small time-steps and follow the evolution of the complex amplitudes in the basis states. In practice even this is impossible for more than a handful of atoms, when you treat all degrees of freedom quantum mechanically. However, the mathematical complexity of solving the time-dependent Schrödinger equation for most molecular systems makes it impossible to obtain exact analytical solutions. We are thus forced to seek numerical solutions based on perturbation or approximation methods that will reduce the complexity. Among these methods, time-dependent perturbation theory is the most widely used approach for calculations in spectroscopy, relaxation, and other rate processes. In this section we will work on classifying approximation methods and work out the details of time-dependent perturbation theory. 2.1. Time-Evolution Operator Let’s start at the beginning by obtaining the equation of motion that describes the wavefunction and its time evolution through the time propagator. We are seeking equations of motion for quantum systems that are equivalent to Newton’s—or more accurately Hamilton’s—equations for classical systems. The question is, if we know the wavefunction at time t0 rt, 0 , how does it change with time? How do we determine rt, for some later time tt 0 ? We will use our intuition here, based largely on correspondence to classical mechanics).
    [Show full text]
  • Lecture 6: Time Evolution and the Schrödinger Equation
    8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology 2013 February 26 Lecture 6 Time Evolution and the Schr¨odinger Equation Assigned Reading: E&R 3all, 51;3;4;6 Li. 25−8, 31−3 Ga. 2all6=4 Sh. 3, 4 The Schr¨odinger equation is a partial differential equation. For instance, if pˆ2 m!2xˆ2 Eˆ = + ; 2m 2 then the Schr¨odinger equation becomes @ 2 @2 m!2x2 i = − ~ + : ~ @t 2m @x2 2 Of course, Eˆ depends on the system, and the Schr¨odinger equation changes accordingly. To fully solve this for a given Eˆ, there are a few different methods available, and those are through brute force, extreme cleverness, and numerical calculation. An elegant way that helps in all cases though makes use of superposition. Suppose that at t = 0, our system is in a state of definite energy. This means that (x; 0; E) = φ(x; E); where Eφˆ (x; E) = E · φ(x; E): Evolving it in time means that @ i E = E · : ~ @t E Given the initial condition, this is easily solved to be (x; t; E) = e −i!tφ(x; E) through the de Broglie relation E = ~!: Note that p(x; t) = j (x; t; E)j2 = jφ(x; E)j2 ; 2 8.04: Lecture 6 so the time evolution disappears from the probability density! That is why wavefunctions corresponding to states of definite energy are also called stationary states. So are all systems in stationary states? Well, probabilities generally evolve in time, so that cannot be. Then are any systems in stationary states? Well, nothing is eternal, and like the plane wave, the stationary state is only an approximation.
    [Show full text]
  • Advanced Quantum Theory AMATH473/673, PHYS454
    Advanced Quantum Theory AMATH473/673, PHYS454 Achim Kempf Department of Applied Mathematics University of Waterloo Canada c Achim Kempf, October 2016 (Please do not copy: textbook in progress) 2 Contents 1 A brief history of quantum theory 9 1.1 The classical period . 9 1.2 Planck and the \Ultraviolet Catastrophe" . 9 1.3 Discovery of h ................................ 10 1.4 Mounting evidence for the fundamental importance of h . 11 1.5 The discovery of quantum theory . 11 1.6 Relativistic quantum mechanics . 12 1.7 Quantum field theory . 14 1.8 Beyond quantum field theory? . 16 1.9 Experiment and theory . 18 2 Classical mechanics in Hamiltonian form 19 2.1 Newton's laws for classical mechanics cannot be upgraded . 19 2.2 Levels of abstraction . 20 2.3 Classical mechanics in Hamiltonian formulation . 21 2.3.1 The energy function H contains all information . 21 2.3.2 The Poisson bracket . 23 2.3.3 The Hamilton equations . 25 2.3.4 Symmetries and Conservation laws . 27 2.3.5 A representation of the Poisson bracket . 29 2.4 Summary: The laws of classical mechanics . 30 2.5 Classical field theory . 31 3 Quantum mechanics in Hamiltonian form 33 3.1 Reconsidering the nature of observables . 34 3.2 The canonical commutation relations . 35 3.3 From the Hamiltonian to the Equations of Motion . 38 3.4 From the Hamiltonian to predictions of numbers . 41 3.4.1 A matrix representation . 42 3.4.2 Solving the matrix differential equations . 44 3.4.3 From matrix-valued functions to number predictions .
    [Show full text]
  • The Time Evolution of a Wave Function
    The Time Evolution of a Wave Function ² A \system" refers to an electron in a potential energy well, e.g., an electron in a one-dimensional in¯nite square well. The system is speci¯ed by a given Hamiltonian. ² Assume all systems are isolated. ² Assume all systems have a time-independent Hamiltonian operator H^ . ² TISE and TDSE are abbreviations for the time-independent SchrÄodingerequation and the time- dependent SchrÄodingerequation, respectively. P ² The symbol in all questions denotes a sum over a complete set of states. PART A ² Information for questions (I)-(VI) In the following questions, an electron is in a one-dimensional in¯nite square well of width L. (The q 2 n2¼2¹h2 stationary states are Ãn(x) = L sin(n¼x=L), and the allowed energies are En = 2mL2 , where n = 1; 2; 3:::) p (I) Suppose the wave function for an electron at time t = 0 is given by Ã(x; 0) = 2=L sin(5¼x=L). Which one of the following is the wave function at time t? q 2 (a) Ã(x; t) = sin(5¼x=L) cos(E5t=¹h) qL 2 ¡iE5t=¹h (b) Ã(x; t) = L sin(5¼x=L) e (c) Both (a) and (b) above are appropriate ways to write the wave function. (d) None of the above. q 2 (II) The wave function for an electron at time t = 0 is given by Ã(x; 0) = L sin(5¼x=L). Which one of the following is true about the probability density, jÃ(x; t)j2, after time t? 2 2 2 2 (a) jÃ(x; t)j = L sin (5¼x=L) cos (E5t=¹h).
    [Show full text]
  • Time Evolution Operator
    Time evolution operator external constant magnetic field parallel to the z-axis, it In quantum mechanics will precess in the xy-plane: the probability for the result ¯h/2 in the measurement SGxˆ oscillates between 0 and 1 • unlike position, time is not an observable. as a function of time. In any case, the probability for the • there is no Hermitean operator whose eigenvalues result ¯h/2 or −¯h/2 remains always as the constant 1. were the time of the system. Generalizing, it is natural to require that X 2 X 2 • time appears only as a parameter, not as a |ca0 (t0)| = |ca0 (t)| . measurable quantity. a0 a0 So, contradictory to teachings of the relativity theory, In other words, the normalization of the states does not time and position are not on equal standing. In depend on time: relativistic quantum field theories the equality is restored hα, t |α, t i = hα, t ; t|α, t ; ti by degrading also the position down to the parameter 0 0 0 0 † level. = hα, t0|U (t, t0)U(t, t0)|α, t0i. We consider a system which at the moment t is in the 0 This is satisfied if we require U(t, t ) to be unitary, i.e. state |αi. When time goes on there is no reason to expect 0 † it to remain in this state. We suppose that at a later U (t, t0)U(t, t0) = 1. moment t the system is described by the state |α, t0; ti, (t > t0), 2. Composition property The evolution from the time t0 to a later time t2 should where the parameter t0 reminds us that exactly at that be equivalent to the evolution from the initial time t0 to moment the system was in the state |αi.
    [Show full text]
  • 1 Observables 2 Schrödinger's Equation
    Observables, Hamiltonian, Schrodinger¨ Equation 1 Observables An observable is a physical quantity, such as energy, spin or position, that can be measured; think of a measuring device with a pointer from which you can read off a real number which is the outcome of the measurement. For a k-state quantum system, observables correspond to k×k hermitian matrices. Recall that † a matrix M is hermitian iff M = M. Since M is hermitian, it has an orthonormal set of eigenvectors φ j with real eigenvalues λ j. If the quantum state of the system is ψ , what is the outcome of the measurement M? To understand this, let us write ψ = a0φ0 + ··· + ak−1 φk− 1 in the { φ j }-basis. Now, the result of 2 the measurement is λ j (this is real number we read off our measurement device) with probability |a j| . Moreover, the state of the system is reset to φ j . It should be clear how this description of a measurement corresponds to what we described earlier while expaining the measurement principle: there a measurement was specified by picking an orthonormal basis 2 { φ j }, and the measurement outcome was j with probability |a j| . The sequence of real numbers λ j simply provide a way of specifying what the pointer of the measurement device indicates for the j-th outcome. Moreover given any orthonormal basis φ j and the sequence of real numbers λ j, we can reconstruct a hermitian matrix M as: M = ∑k−1 λ φ φ ; in the { φ }-basis this is just a diagonal matrix with the λ s j=0 j j j j j on the diagonal.
    [Show full text]