Nonlinear Control Seminar
Quantum Mechanics with Control Theory

Gregory J. Toussaint

Date: 11 August 1999

• H. H Rosenbrock has proposed a new application for dynamic programming [8].
• Review dynamic programming and minimum principle (briefly).
• Review quantum mechanics (not so briefly).
• Use dynamic programming to develop quantum mechanics.
• Some of Rosenbrock's philosophy.

Given a system described by a differential equation:

$$\dot{x} = f(x,u,t), \hspace{2em} x(0) = x_0$$

and a cost function to be optimized

$$J_{[t,t_f]} \, (u) = q(t_f, x(t_f)) + \int_t^{t_f} g(x(s), u(s), s)\, ds$$

define the optimal cost as

$$V(t,x) := \min_{u_{[t,t_f]}} J_{[t,t_f]} \, (u).$$

Dynamic programming leads to an HJB equation:

$$\begin{eqnarray*}- V_t(t,x) & = & \min_{u \in \mathcal{U}} \left\{ V_x(t,x) \cdot f(x,u,t) + g(x,u,t) \right\} \\ V(t_f, x) & = & q(t_f, x). \end{eqnarray*}$$

We could also use a Lagrangian to derive the minimum principle. Let H be the Hamiltonian and $\ensuremath{p^{T}}$ be the co-state vector.

$$\begin{eqnarray*}H & = & g(x,u,t) + \ensuremath{p^{T}}\, f(x,u,t) \\ \dot{p} & ... ...} & = & f(x,u,t) \;\;=\;\; H_{p} \, , \hspace{2em} x(t_0) = x_0. \end{eqnarray*}$$

• The standard development of quantum mechanics is based on a set of postulates [1].
• Postulate 1: For any possible state of a system, there is a function, $\Psi$, of the coordinates of the parts of the system and time that completely describes the system.
• For example, a single particle with Cartesian coordinates
  $\Psi = \Psi(x,y,z,t)$.
• A general system uses generalized coordinates, q_i, with
  $\Psi = \Psi(q_i, t)$.
• Recall there is a duality between particles and waves. For electromagnetic radiation we have
  
  $$E = h \nu = \frac{hc}{\lambda}.$$
  
  
  We also know E = mc² for a photon. So we get
  
  $$\begin{eqnarray*}m c^2 & = & \frac{hc}{\lambda} \\ \lambda & = & \frac{h}{mc}. \end{eqnarray*}$$
  
  De Broglie: if a particle had wave properties, we would have
  
  $$\lambda = \frac{h}{mv}.$$
  
  

• The model is of a wave, so $\Psi$ is called a wave function.
• The state of the system is the quantum state.

• $\Psi^2$ is proportional to probability.
• $\Psi$ may be complex, so consider $\Psi^* \Psi$.
• $\Psi^*\Psi \, d \tau$ is proportional to probability of finding the particles in volume element $d \tau = dx\,dy\,dz$.
• Particles must be somewhere so
  
  $$\int_{X} \Psi^*\Psi \, d \tau = 1$$
  
  
  which means $\Psi$ is normalized.
• For $\Psi$ to be well-behaved, we also require it to be finite, single-valued and continuous.
• Recall the classical three dimensional wave equation
  
  $$\ensuremath{\frac{\partial^2 \phi}{\partial x^2}} + \ensurema... ...frac{1}{v^2}\,\ensuremath{\frac{\partial^2 \phi}{\partial t^2}}$$
  
  
  where $\phi$ is an amplitude function and v is the phase velocity of the wave.
• For harmonic motion (sine wave) we get
  
  $$\ensuremath{\frac{\partial^2 \phi}{\partial t^2}} = -4 \pi^2 \nu^2 \phi$$
  
  
  We recall $\lambda = \frac{h}{mv}$ from De Broglie's observations and we know $v = \lambda \nu$ or $\lambda = \frac{v}{\nu}$. Rearrange to get:
  
  $$\frac{v}{\nu} = \frac{h}{mv} \hspace{1em} \Rightarrow \hspace{1em} \nu^2 = \frac{m^2 v^4}{h^2}.$$
  
  
  Substitute to get
  
  $$\ensuremath{\frac{\partial^2 \phi}{\partial t^2}} = -4 \pi^2 \left( \frac{m^2 v^4}{h^2} \right) \phi$$
  
  
  
  
  $$\Rightarrow \;\; \ensuremath{\frac{\partial^2 \phi}{\partial ... ...\frac{1}{v^2} \left( \frac{-4 \pi^2 v^4 m^2}{h^2} \right) \phi$$
  
  

• Define:

  E	=	Total energy
  T	=	Kinetic energy
  V	=	Potential energy

  where $T = \frac{m v^2}{2} = E - V$.

  Substitute to get:
  

  $$\ensuremath{\frac{\partial^2 \phi}{\partial x^2}} + \ensurema... ...\frac{-4 \pi^2 m^2}{h^2} \left[ \frac{2(E-V)}{m} \right] \phi.$$
  
  
  Rearrange:
  
  $$\ensuremath{\frac{\partial^2 \phi}{\partial x^2}} + \ensurema... ...^2 \phi}{\partial z^2}} + \frac{8 \pi^2 m}{h^2} (E-V) \phi = 0$$
  
  

• This is one form of Schrödinger's wave equation. The solutions are wave functions.

• Postulate 2: For every dynamical variable (classical observable) there is a corresponding operator.
• Dynamic variables include energy, momentum, angular momentum, and position coordinates.
• These variables are replaced by operators such as $( \cdot )^2$, $\ensuremath{\frac{\partial }{\partial x}}$, or $\int$.
• The coordinates are the same as the operators.
• Operators are linear and Hermitian:
  
  $$\begin{eqnarray*}\alpha(\phi_1 + \phi_2) & = & \alpha \phi_1 + \alpha \phi_2 \\ ... ...a \phi_2 \, d \tau & = & \int \phi_2 \alpha^* \phi_1^* \, d \tau \end{eqnarray*}$$
  
  
• Linear nature is related to superposition of states and waves reinforcing each other.

 

Table: Operators used in quantum mechanics.

Quantity	Symbol		Operator Form
Momentum x	px		$\frac{\hbar}{i} \ensuremath{\frac{\partial }{\partial x}}$
Kinetic Energy	$\frac{p^2}{2m}$		$- \frac{\hbar^2}{2m} \left[ \ensuremath{\frac{\partial^2 }{\partial x^2}} + \en... ...rtial^2 }{\partial y^2}} + \ensuremath{\frac{\partial^2 }{\partial z^2}}\right]$
Kinetic Energy	T		$-\frac{\hbar}{i} \ensuremath{\frac{\partial }{\partial t}}$
Potential Energy	V		V(qi)

• Postulate 3: The permissible values that a dynamical variable may have are those given by $\alpha \phi = a \phi$, where $\phi$ is the eigenfunction of the operator $\alpha$ that corresponds to the observable whose permissible values are a.
• Since the properties vary, we want to determine the expected value. Given the operator equation
  
  $$\alpha \phi = a \phi$$
  
  
  multiply by $\phi^*$
  
  $$\phi^* \alpha \phi = \phi^* a \phi = a \phi^* \phi$$
  
  
  and integrate to get
  
  $$\int_X \phi^* \alpha \phi \, d \tau = \int_X a \, \phi^* \phi \, d \tau$$
  
  
  
  
  $$\Rightarrow \;\; \bar{a} = \langle a \rangle = \frac{\int_X \... ... \phi^* \phi \, d \tau} = \int_X \phi^* \alpha \phi \, d \tau$$
  
  
  if $\phi$ is normalized.
• If the wave function is known, then we can find the expected value for a given parameter using its operator.

• Postulate 4: The state function, $\Psi$, is given as a solution of $\Bbb{H} \, \Psi = E \Psi$, where $\Bbb{H}$ is the operator for total energy, the Hamiltonian operator.
• This postulate is the starting point for formulating a problem in quantum mechanical terms.
• The Hamiltonian in classical physics is the total energy, H = T + V or in operator form $\Bbb{H} = \Bbb{T} + \Bbb{V}$.
• Written in generalized coordinates, q_i, and time, the starting equation becomes
  
  $$\Bbb{H} \, \Psi (q_i, t) = - \frac{\hbar}{i} \ensuremath{\frac{\partial \Psi(q_i,t)}{\partial t}}$$
  
  

• Kinetic energy: $T = \frac{m v^2}{2} = \frac{p^2}{2m}$. In operator form we get:
  
  $$\Bbb{T} = - \frac{\hbar^2}{2m} \left( \ensuremath{\frac{\part... ...al^2 }{\partial z^2}}\right) =: - \frac{\hbar^2}{2m} \nabla^2$$
  
  

• Potential energy: V = V(q_i, t).
• Total energy operator equation:
  
  $$\left[ - \frac{\hbar^2}{2m} \nabla^2 + \Bbb{V}(q_i, t) \right... ...{\hbar}{i} \ensuremath{\frac{\partial \Psi(q_i,t)}{\partial t}}$$
  
  
  which is Schrödinger's time-dependent equation.

• If we assume we can separate the variables
  
  $$\Psi (q_i, t) = \psi(q_i) \, \tau(t)$$
  
  

• Then using
  
  $$\Bbb{H} \, \Psi(q_i,t) = \Bbb{H} \, \psi(q_i) \tau(t)$$
  
  
  we can show that
  
  $$\begin{eqnarray*}\tau(t) & = & e^{-i W t / \hbar} \\ \Psi(q_i, t) & = & \psi(q_i) \, e^{-i W t / \hbar} \end{eqnarray*}$$
  
  and we get Schrödinger's time-independent equation:
  
  $$\Bbb{H} \, \psi = E \psi$$
  
  

• This is the standard approach to developing quantum mechanics.
• Rosenbrock suggests there is a better way using dynamic programming principles.

• Hamilton's principle gives a concise and elegant basis for classical mechanics.
• Bellman noted that Hamilton's principle falls within the scope of dynamic programming.
• Consider the application for a single particle. (A more general description is given in [2].)
• If H(x,p,t) is the Hamiltonian (total energy) of a particle moving from x at t to x_f at t_f, then we can state Hamilton's principle as
  

  $$\underset{v,\,p}{\delta}\int_{t}^{t_f} \left[ p v - H (x,p,\tau) \right] \, d \tau = 0, \hspace{1em} v = \frac{dx}{dt}.$$ (1)

  
  
  Note that pv is two times the kinetic energy and H is the total energy, so the quantity in square brackets is the Lagrangian, L = T - V.
• After we optimize, p and v become functions of x and t.
• Write W(x,t) for the optimal (stationary) value of the integral, which is a cost.
• We can now write Bellman's equation:
  

  $$\underset{v,\,p}{\mathrm{stat}}\left[ \frac{d W}{d t} + pv -... ...ensuremath{\frac{\partial W}{\partial x}} + pv - H \right] = 0$$ (2)

  
  
  which gives the following
  

  $$p = - \ensuremath{\frac{\partial W}{\partial x}} , \hspace{1... ... \hspace{1em} H = \ensuremath{\frac{\partial W}{\partial t}} .$$ (3)

  
  

• Take the complete time derivative of p to get
  
   

  $$\frac{d \,p (x,t) }{dt} \ensuremath{\frac{\partial p}{\partial t}} + \ensuremath{\frac{\partial p}{\partial x}}\,\ensuremath{\frac{\partial x}{\partial t}}$$ =

  $$- \left[ \ensuremath{\frac{\partial^2 W}{\partial x \partial t}} + v \ensuremath{\frac{\partial^2 W}{\partial x^2}}\right]$$ =

  $$- \left[ \ensuremath{\frac{\partial H}{\partial x}} - \ensuremath... ...ac{\partial H}{\partial p}}\, \ensuremath{\frac{\partial p}{\partial x}}\right]$$ = (4)

  
  

• If in H(x,p,t), x and p are independent variables, then (3) and (4) give Hamilton's equations
  

  $$\frac{dx}{dt} = \ensuremath{\frac{\partial H}{\partial p}} ,... ...em} \frac{dp}{dt} = -\ensuremath{\frac{\partial H}{\partial x}}$$ (5)

  
  

• Equations in (5) look like the minimum principle.
• Example with a particle moving in a gravitational field.
• A valuable application of dynamic programming, but nothing new.

• Quantum mechanics considers complex functions such as $\psi$ and probability distributions of variables.
• To introduce these new features, replace x by complex variable q and replace $\; v = \frac{dx}{dt} \;$ with $\; dq = \nu d \tau + n dz \,$, where z is a normalized Wiener process and n is complex with $n^2 = - i \hbar/m$.
• New statement of Hamilton's principle:
  

  $$\underset{\tilde{v}, \, \tilde{p}}{\delta}\, E \left\{ \int_... ... \tilde{H} (q, \tilde{p}, \tau) \right] \, d \tau \right\} = 0$$ (6)

  
  
  where the tilde indicates a complex variable.
• Bellman's equation becomes
  

  $$\underset{\tilde{v}, \, \tilde{p}}{\mathrm{stat}}\; E \left[... ...tilde{p} \tilde{v} - \tilde{H}(q, \tilde{p}, \tau) \right] = 0$$ (7)

  
  
  where we use
   $$\begin{gather} E [dz] = 0, \hspace{1em} E[dz^2] = dt, \nonumber \\ \hspace{1em}... ...[\tilde{v}^2 dt^2 + 2 \tilde{v} dt \, n dz + n^2 dz^2 ] = n^2 \, dt \end{gather}$$
  to the first order in dt.
• Using a similar approach as with the classical case, we now get
  

  $$\tilde{p} = - \ensuremath{\frac{\partial \tilde{W}}{\partial... ...2m}\, \ensuremath{\frac{\partial^2 \tilde{W}}{\partial q^2}} .$$ (8)

  
  

• So far we've made the variables in (3) complex and added a second order term to the last equation to account for the stochastic nature of the problem.
• Now make a change of variables:
  
  $$\begin{eqnarray*}\tilde{W} & = & i \hbar \log \psi \\ \hat{p} & = & - i \hbar \ensuremath{\frac{\partial }{\partial q}}\end{eqnarray*}$$
  
  where $\psi$ is a function of q and t and $\hat{p}$ is an operator for momentum.
• We can then show the following:
    $$\begin{gather}\tilde{p} = - \ensuremath{\frac{\partial \tilde{W}}{\partial q}} =... ... t}} = \left( \frac{\hat{p}^2}{2m} + V \right) \psi = \hat{H} \psi. \end{gather}$$
  
• Equation (11) is Schrödinger's equation extended from the real line to the complex plane designated by q.
• Standard quantum mechanics looks at complex functions on a real space. Now we have complex functions on a complex space.
• To get the standard theory, make the following postulate:

  If the expected value of some property (such as momentum $\tilde{p}$or energy $\tilde{H}$) of the complex images in an ensemble has the real value $\alpha$ on the real axis, then $\alpha$ is the value of the corresponding property of the physical particles.

• If $\alpha$ is a real constant and we have $\tilde{H} = \psi^{-1} \hat{H} \psi = \alpha$, then $\hat{H} \psi = \alpha \psi$, which can only be true if $\alpha$ is an eigenvalue of operator $\hat{H}$ and $\psi$ is the corresponding eigenvector.

• Equation (6) and a single postulate allow us to get results usually given as separate postulates in the elementary development of quantum mechanics:

  1.

  Momentum p in classical mechanics is to be replaced on the real axis by $\hat{p} = -i \hbar \ensuremath{\frac{\partial }{\partial x}}$.

  2.

  The function $\exp [ \tilde{W}/ i \hbar]$ is the wave function $\psi$, satisfying Schrödinger's equation.

  3.

  The density of particles in an ensemble on the real axis is $\rho = \psi^* \psi$.

  4.

  The density satisfies
  

  $$\ensuremath{\frac{\partial \rho}{\partial t}} + \ensuremath{\frac{\partial \rho u}{\partial x}} = 0$$ (9)

  
  

  where $u = \Re \{ \tilde{v} \}$ which is $\Re \{ \psi^{-1} \hat{p} \psi / m \}$ for a single particle.

  5.

  When $\psi$ corresponds to a real quantity such as $\tilde{p}$or $\tilde{H}$, it must be an eigenfunction of the corresponding operator $\hat{p}$ or $\hat{H}$. The corresponding eigenvalue is the value of the variable.

  6.

  When a system is to be considered as partly classical and partly quantum mechanical, a particular symmetrical combination of the variables is to be chosen.

• Approach provides a compact basis for quantum mechanics and offers a direct link to classical mechanics.
• Approach also helps explain some results which are obscure in the standard development:

  1.

  Feynman's path integral method gives the phase of $\psi$correctly, but not the dependence of $\vert\psi\vert$ on time. The method appears to not account for a delay contribution to $\psi$ from complex images not following a real trajectory. See Figure 1.

  2.

  Interference in the two-slit experiment can be explained by a change in the ensemble for the expectations in equation (6) [3].

  3.

  In an electromagnetic field, the field strength can be shown to have singularities with the features of photons [7].

  4.

  The introduction of random variable z in (6) determines the direction of time, which is also implied by dynamic programming.

  

Figure: Dynamic programming with complex trajectories.

• A deeper point is that given appropriate end conditions, equation (6) can have solutions of two different kinds:

  1.

  Open-loop: gives optimal velocity as a function of time alone. We follow a schedule of velocities regardless of perturbations.

  2.

  Closed-loop: optimal velocity is a function of q and time. This approach continually re-evaluates the velocities to get the optimal cost from the perturbed position.

• Closed-loop control implies a goal-seeking behavior.
• Quantum-mechanics can be obtained from a closed-loop solution of (6) and not from an open-loop solution
• The goal the system seeks to satisfy is to make the expectation of the integral stationary.
• We can describe the results with or without reference to the underlying purpose, because every feasible purpose has a policy that will accomplish it.
• Purpose or lack of purpose are not properties of the system, but are properties of the description we choose.
• If the policy accurately reflects the purpose, there is no way to distinguish between the two.

Consider:

• Living organisms are made of elementary particles such as protons, neutrons, and electrons.
• Standard scientific theories propose that elementary particles are inert and purposeless and evolve according to causal laws.
• Living organisms can exhibit clear purposes.
• Question: How can purpose arise from a purposeless substrate?
• Standard Answer: Natural selection acts on chance variations to produce this effect.
• If we describe quantum mechanics as goal-seeking, the situation is different.
• The above argument answers the wrong question.
• The correct question: How can a new purpose arise in living organisms from a substrate which already has purpose, given that an organism cannot simultaneously seek two goals unless these are in a special relationship with each other?
• Another question: What is this special relationship?