Quantum mechanics on continuous Hilbert spaces

One of the more physically meaningful classes of Hilbert spaces are those which are continuous. Such systems typically describe the dynamics of quantum particles propagating in some physical continuum, that is, \(\Reals^\Dimension\) for any \(\Dimension \in \IntegersPositive\). Mathematically, any continuous quantum system may be described by a state, which is a vector \(\ket{\WaveFunction}\) residing in a (continuous) Lebesgue space (a special function space that is also a type of Hilbert space), denoted by \(\SpaceLebesgue^2(\SetSub)\), on certain set \(\SetSub\) which is some configuration space (such as \(\Reals^3)\). The theory discussed here adapts the work presented in [2, 4, 12, 4, 5, 6, 7, 8, 9, 10].

Probabilities and the Born rule

For a (measurable) function \(\WaveFunction\), the condition \(\WaveFunction \in \SpaceLebesgue^2(\SetSub)\) specifies that \(\WaveFunction\) satisfies a finitely bound integral of the form

(197)\[\norm{\WaveFunction}_2 \equiv \left[\int_{\SetSub}\abs{\WaveFunction(\vec{\Coordinate})}^2 \, \diff{\Measure(\vec{\Coordinate})}\right]^{\frac{1}{2}} < \infty\]

where \(\norm{\WaveFunction}_2\) is the \(\SpaceLebesgue^2\)-norm of \(\WaveFunction\) and \(\Measure(\vec{\Coordinate})\) is the appropriate measure of the set \(\SetSub\). A function for which (197) holds true is said to be square-integrable (on \(\SetSub\)), or an \(\SpaceLebesgue^2\)-function. Note that such functions form an inner product space with the inner product given by

(198)\[\braket{\phi}{\WaveFunction} = \int_{\SetSub} \conj{\phi}{(\vec{\Coordinate})}\WaveFunction(\vec{\Coordinate})\,\diff{\Measure(\vec{\Coordinate})}, \qquad \phi,\WaveFunction\in \SpaceLebesgue^2(\vec{\Coordinate}).\]

With this, the Lebesgue space \(\SpaceLebesgue^2(\SetSub)\) is now a Hilbert space \(\SpaceHilbert\), and we have the isomorphism \(\SpaceLebesgue^2 \cong \SpaceHilbert\). From here, if the norm defined in (197) is equal to \(1\),

(199)\[\int_{\SetSub}\abs{\WaveFunction(\vec{\Coordinate})}^2 \, \diff{\Measure(\vec{\Coordinate})} = 1,\]

then the element \(\WaveFunction\in \SpaceLebesgue^2(\SetSub)\) defines a probability measure on \(\SetSub\). In the case that the measure \(\Measure(\vec{\Coordinate})\) is continuous, then the real-valued function

(200)\[\Probability(\vec{\Coordinate}) = \abs{\WaveFunction(\vec{\Coordinate})}^2\]

is called a probability density function (PDF) and the complex-valued function \(\WaveFunction(\vec{\Coordinate})\) is called a probability amplitude. If the measure \(\Measure(\vec{\Coordinate})\) is instead discrete, then the integral becomes a sum and \(\abs{\WaveFunction(\vec{\Coordinate})}^2\) defines the probability measure on the set \(\SetSub\)—that is, the probability that the quantum system represented by \(\ket{\WaveFunction}\) assumes the state with parameter \(\vec{\Coordinate} \in \SetSub\). The form (200) is perhaps the archetypal representation of the Born rule for continuous quantum mechanics.

The state vector \(\ket{\WaveFunction}\) may be projected into different coordinate bases via the equivalence

(201)\[\WaveFunction(\vec{\Coordinate}) = \braket{\vec{\Coordinate}}{\WaveFunction} \quad \Longleftrightarrow \quad \ket{\WaveFunction} = \int \diff{\Measure(\vec{\Coordinate})} \, \WaveFunction(\vec{\Coordinate}) \ket{\vec{\Coordinate}},\]

where the configuration of the associated system in \(\Dimension\)-dimensional space is represented by the \(\Dimension\)-tuple of generalized coordinates \((\Position_j)=\vec{\Position}\). Any such projection onto any basis is, by the usual interpretation of quantum mechanics, termed the wave function of the associated system. Commonly, this projection may be generalized to incorporate the state’s time dependence,

(202)\[\WaveFunction(\vec{\Position},\Time) = \braket{\vec{\Position},\Time}{\WaveFunction},\]

which we interpret as the probability amplitude that the associated system (e.g., a particle) has position \(\vec{\Position}\) at time \(\Time\). Accordingly, the PDF is given by the Born rule (200),

(203)\[\Probability(\vec{\Position},\Time) = \abs{\WaveFunction(\vec{\Position},\Time)}^2,\]

which describes the probability of the particle’s presence at and around the point \((\vec{\Position},\Time)\).

Time evolution

Suppose we have some (closed) system which is described by the state \(\ket{\WaveFunction(\TimeInitial)}\) defined on a complex Hilbert space at some initial time \(\TimeInitial\). By (176), the time evolution of this system over the interval \(\TimeInitial \rightarrow \TimeFinal\) to some final state \(\ket{\WaveFunction(\TimeFinal)}\) is described by the form

(204)\[\ket{\WaveFunction(\TimeFinal)} = \UnitaryTime(\TimeFinal,\TimeInitial)\ket{\WaveFunction(\TimeInitial)}.\]

Given that \(\WaveFunction(\Time)\) must remain normalized for all \(\Time\), then the operator \(\UnitaryTime\) must therefore be a unitary transformation. Importantly, this means that the transformation from past to future must be:

1. invertible \(\implies\) information is conserved,

2. norm-preserving \(\implies\) probability is conserved.

These facts imply that \(\UnitaryTime\) may be expressed in general as the exponential map

(205)\[\UnitaryTime(\TimeFinal,\TimeInitial) = \mathcal{T} \exp\left[-\frac{\eye}{\hbar} \int_{\TimeInitial}^{\TimeFinal} \OperatorHamiltonian(\Time) \, \diff{\Time}\right]\]

where, given that the Lie algebra of the unitary group is generated by Hermitian operators, then \(\OperatorHamiltonian\) is a Hermitian operator, i.e., \(\OperatorHamiltonian^\dagger = \OperatorHamiltonian\). Here, the time-ordering symbol \(\mathcal{T}\) permutes a product of non-commuting operators \({\op{A}_n}\) into the uniquely determined reordered expression

(206)\[\mathcal{T}[\op{A}_1(\Time_1)\cdot \op{A}_2(\Time_2)\cdot\ldots\cdot \op{A}_n(\Time_n)] = \op{A}_{i_1}(\Time_{i_1})\cdot \op{A}_{i_2}(\Time_{i_2})\cdot\ldots\cdot \op{A}_{i_n}(\Time_{i_n}), \qquad \Time_{i_1}\geq \Time_{i_2}\geq\ldots\geq \Time_{i_n}.\]

This result is a causal chain where the primary cause in the past is \(\op{A}_{i_n}(\Time_{i_n})\) while the future effect is \(\op{A}_{i_1}(\Time_{i_1})\). In the case where \(\OperatorHamiltonian\) is time-independent, the exponential map is simply

(207)\[\UnitaryTime(\TimeFinal,\TimeInitial) = \exp\left[-\frac{\eye}{\hbar} (\TimeFinal-\TimeInitial)\OperatorHamiltonian \right].\]

Substituting this form into equation (204), letting \((\TimeFinal,\TimeInitial)\rightarrow(\Time,0)\) and then differentiating both sides with respect to \(\Time\) yields

(208)\[\OperatorHamiltonian\ket{\WaveFunction(\Time)} = \eye\hbar\frac{\partial}{\partial \Time}\ket{\WaveFunction(\Time)}.\]

This is the time-dependent Schrödinger equation (TDSE), which is perhaps one of the most fundamental and important equations of non-relativistic quantum mechanics. Here, we identify the operator \(\OperatorHamiltonian\) as the (time-independent) Hamiltonian operator, which corresponds to (or, more formally, has eigenvalues of) the total energy \(\Energy\) of the state \(\WaveFunction\), i.e., \(\OperatorHamiltonian\ket{\WaveFunction} \equiv \Energy\ket{\WaveFunction}\). Schrödinger’s equation is a partial differential equation that governs the temporal evolution of any time-dependent quantum system described by the arbitrary state \(\ket{\WaveFunction(\Time)}\). Essentially, it is the equation of motion which all non-relativistic systems must obey in order to be physical.

For any time-independent system, like a particle with (classical) kinetic energy \(\Kinetic\) in a static potential field \(\Potential\), the Hamiltonian may be written as the sum of the (time-independent) total kinetic \(\OperatorKinetic\) and potential \(\OperatorPotential\) (operator) energies,

(209)\[\OperatorHamiltonian(\vec{\Position},\vec{\Momentum}) = \OperatorKinetic(\vec{\Position},\vec{\Momentum}) + \OperatorPotential(\vec{\Position})\]

where \(\vec{\Momentum}\equiv(\Momentum_j)\) represents the \(\Dimension\)-tuple of generalized momenta. Consequently, the TDSE is the mathematical statement that the time-independent Hamiltonian \(\OperatorHamiltonian\) (209) is the infinitesimal generator of a one-parameter group parametrized by the time \(\Time\). This is to say that the time evolution of a state \(\ket{\WaveFunction(\TimeInitial)}\) to a later state \(\ket{\WaveFunction(\TimeFinal)}\) is generated by the Hamiltonian.

The time-evolution operator’s utility can be observed in many different contexts. For example, consider an arbitrary single-body system located at position \(\vec{\Position}\) at time \(\Time\). The state (or more specifically, the location) of such a system can be defined in the \(\vec{\Position}\)-space (configuration) basis as

(210)\[\ket{\vec{\Position},\Time} \equiv \UnitaryTime(\Time,0)\ket{\vec{\Position}},\]

so that we may write the overlap between the past \(\ket{\vec{\PositionInitial},\TimeInitial}\) and future \(\ket{\vec{\PositionFinal},\TimeFinal}\) states as

(211)\[\braket{\vec{\PositionFinal},\TimeFinal}{\vec{\PositionInitial},\TimeInitial} = \bra{\vec{\PositionFinal}}\Unitary(\TimeFinal,\TimeInitial)\ket{\vec{\PositionInitial}},\]

The physical significance of this quantity, known as the transition amplitude, shall next be discussed.

The path-integral formulation

The path-integral formulation [4, 11] (or the sum-over-histories approach) of quantum mechanics and field theory is a formalism that generalizes the principle of stationary action (from classical mechanics) to quantum mechanics. It is based on the notion of the propagator,

(212)\[\Propagator(\vec{\PositionFinal},\TimeFinal;\vec{\PositionInitial},\TimeInitial) \equiv \braket{\vec{\PositionFinal},\TimeFinal}{\vec{\PositionInitial},\TimeInitial}, \qquad \TimeFinal>\TimeInitial,\]

which is, in general, a generalized function (often a distribution), and is defined as per (211). Of its many properties and applications, the primary utility of the propagator lies in its ability to evolve a wave function \(\WaveFunction(\vec{\PositionInitial},\TimeInitial)\) to a later state via convolution, e.g.,

(213)\[\WaveFunction(\vec{\PositionFinal},\TimeFinal) = \int_{\Reals^\Dimension} \Propagator(\vec{\PositionFinal},\TimeFinal;\vec{\PositionInitial},\TimeInitial) \, \WaveFunction(\vec{\PositionInitial},\TimeInitial) \, \diff^\Dimension{\vec{\PositionInitial}}.\]

Here,

(214)\[\WaveFunction(\vec{\Position},\Time) \equiv \braket{\vec{\Position},\Time}{\WaveFunction}\]

is the wave function of the system in state \(\ket{\WaveFunction}\) projected into \(\vec{\Position}\)-space at time \(\Time\). Given both its convolution property in (213) and its relation to the time-evolution operator in (211), we say that the propagator is the Green’s function (or integral kernel) for the Schrödinger equation (208).

Similar to how a wave function can describe the probability amplitude for a particle’s location \(\vec{\Position}\) at time \(\Time\), we may say that the propagator \(\Propagator(\vec{\PositionFinal},\TimeFinal;\vec{\PositionInitial},\TimeInitial)\) is the probability amplitude for the transition of the particle from \((\vec{\PositionInitial},\TimeInitial)\) to \((\vec{\PositionFinal},\TimeFinal)\). In \(\Dimension\)-dimensional space, this transition could be accomplished through an infinite number of distinct and unique trajectories.

In general, the propagator can be written as the phase-space path integral over all paths \(\vec{\Position}(\Time)\) and momentum evolutions \(\vec{\Momentum}(\Time)\) from \((\vec{\PositionInitial},\TimeInitial)\) to \((\vec{\PositionFinal},\TimeFinal)\),

(215)\[\Propagator(\vec{\PositionFinal},\TimeFinal;\vec{\PositionInitial},\TimeInitial) = \int_{\vec{\PositionInitial}}^{\vec{\PositionFinal}} \DifferentialPath\vec{\Position} \, \int_{\vec{\MomentumInitial}}^{\vec{\MomentumFinal}} \frac{\DifferentialPath\vec{\Momentum}}{(2\pi\hbar)^\Dimension} \, \e^{\eye \Action[\vec{\Position}]/\hbar}\]

where \(\Action[\vec{\Position}]\) is the action of path \(\vec{\Position}(\Time)\). Here, we introduced the notation for the position and momentum differentials

(216)\[\begin{aligned} \int_{\vec{\Position}_0}^{\vec{\Position}_\Number} \DifferentialPath\vec{\Position} &\equiv \lim\limits_{\Number \rightarrow \infty} \prod_{k=1}^{\Number-1} \int_{\Reals^\Dimension} \diff^\Dimension{\vec{\Position}_k}, \end{aligned}\]

(217)\[\begin{aligned} \int_{\vec{\Momentum}_0}^{\vec{\Momentum}_\Number} \DifferentialPath\vec{\Momentum} &\equiv \lim\limits_{\Number \rightarrow \infty} \prod_{n=0}^{\Number-1} \int_{\Reals^\Dimension} \diff^\Dimension{\vec{\Momentum}_n}, \end{aligned}\]

which signify integration over all position \(\vec{\Position}(\Time)\) and momentum \(\vec{\Momentum}(\Time)\) evolutions through the \(\Number\) subdivisions of our phase space. This discretization arises from our partitioning of the time coordinate into \(\Number\) “slices”—physically, \(\vec{\Position}_n\) represents the position of the particle at time \(\Time_n\) (where \(n \in \Integers_1^\Number\))—and it is only after taking the continuum limit that we are able to recover smooth dynamics as described by (215).

For a system with a Hamiltonian whose dependence on momentum is purely quadratic, the integral over the momentum evolutions \(\vec{\Momentum}(\Time)\) in (215) can be directly computed. Subsequently, the propagator reduces to the configuration-space path integral,

(218)\[\Propagator(\vec{\PositionFinal},\TimeFinal;\vec{\PositionInitial},\TimeInitial) = \int_{\vec{\PositionInitial}}^{\vec{\PositionFinal}} \DifferentialPath^{\prime}\vec{\Position} \, \e^{\eye \Action[\vec{\Position}]/\hbar},\]

where the “normalized” differential (compared to (216)) is defined as

(219)\[\int_{\vec{\Position}_0}^{\vec{\Position}_\Number} \DifferentialPath^{\prime}\vec{\Position} \equiv \lim\limits_{\Number \rightarrow \infty} \left[\frac{\Mass}{2\pi\eye\hbar(\TimeFinal - \TimeInitial)}\right]^\frac{\Number\Dimension}{2} \prod_{k=1}^{\Number-1} \int_{\Reals^\Dimension} \diff^\Dimension{\vec{\Position}_k}.\]

Here, \(\Mass\) is the system’s mass parameter. Note that this is an integral over all possible paths (both classical and quantum) and not just the classical path \(\classical{\vec{\Position}}(\Time)\) (that which satisfies the relevant classical equations of motion) with action \(\classical{\Action} \equiv \Action[\classical{\vec{\Position}}]\). This expression is often presented as the typical form of the path integral.

References

[1]

E. Zeidler, Quantum Field Theory I: Basics in Mathematics and Physics, Springer, Berlin, Heidelberg, 2006.

[2]

J. Dimock, Quantum Mechanics and Quantum Field Theory: A Mathematical Primer, Cambridge University Press, Cambridge, 2011.

[3]

P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 4th edition, February 1982.

[4] (1,2)

L. H. Ryder, Quantum Field Theory, Cambridge University Press, Cambridge, 2nd edition, 1996.

[5]

A. Zee, Quantum Field Theory in a Nutshell, Princeton University Press, Princeton, N.J, 2nd edition, February 2010.

[6]

J. S. Townsend, A Modern Approach to Quantum Mechanics, University Science Books, Sausalito, California, 2nd edition, February 2012.

[7]

D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, Cambridge University Press, 3rd edition, August 2018.

[8]

K. Schulten, Notes on Quantum Mechanics, CreateSpace Independent Publishing Platform, August 2014.

[9]

W. Dittrich and M. Reuter, Classical and Quantum Dynamics: From Classical Paths to Path Integrals, Springer International Publishing, Cham, 2020.

[10]

L. S. Schulman, Techniques and Applications of Path Integration, Wiley, New York, 1981.

[11]

H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, 5th edition, 2009.