Hamiltonians in Coordinate Space

A Hamiltonian is the operator that generates time evolution. For an autonomous closed system it also represents a conserved total energy; an explicitly time-dependent Hamiltonian, a time-dependent gauge representation, or a non-Hermitian effective Hamiltonian requires that energy statement to be qualified. In coordinate-space wave mechanics, the Hamiltonian usually appears as a differential operator acting on wavefunctions. The practical task is to translate a physical setup into a Hilbert space, coordinates, kinetic-energy operator, potential-energy operator, and domain.

The shortest warning is:

$$\text{Hamiltonian expression} \ne \text{complete quantum problem}.$$

The same differential expression can define different quantum systems when the coordinate domain or boundary conditions change.

Required background. Coordinate Representation supplies the operator representation. Time-Dependent Schrödinger Equation in Coordinate Space supplies the initial-value role of $\hat H$. Time-Independent Schrödinger Equation supplies its stationary spectral role.

Helpful background. Self-Adjoint Operators places the domain and boundary-form checks in their general operator-theoretic setting.

From Classical Energy To Operator

For a single nonrelativistic particle in one dimension, the classical energy often has the form

$$H(x,p)=\frac{p^2}{2m}+V(x).$$

In the position representation,

$$\hat x\psi(x)=x\psi(x), \qquad \hat p\psi(x)=-i\hbar\frac{d\psi}{dx}.$$

This gives the coordinate-space Hamiltonian

$$\hat H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} +V(x).$$

The time-dependent Schrödinger equation is then

$$i\hbar\frac{\partial\psi(x,t)}{\partial t} = \left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} +V(x,t) \right]\psi(x,t),$$

when the potential may depend on time. For time-independent $V(x)$, the stationary equation is

$$\left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} +V(x) \right]\psi(x) =E\psi(x).$$

This construction is not blind substitution. The displayed rule assumes a constant mass, Cartesian coordinates, and a local scalar potential with no ordering ambiguity. More general classical expressions can lead to operator-ordering, constraint, and measure questions that must be settled by the physical model rather than by the replacement $p\mapsto-i\hbar\partial_x$ alone.

Kinetic Energy

On the full line with ordinary Cartesian coordinate $x$, the kinetic-energy operator is

$$\hat T = \frac{\hat p^2}{2m} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.$$

In three-dimensional Cartesian coordinates,

$$\hat T = -\frac{\hbar^2}{2m}\nabla^2.$$

The geometric Laplacian is coordinate invariant, but its coordinate expression and the integration measure change. For a metric $g_{ij}$,

$$\Delta_g = \frac{1}{\sqrt g} \partial_i\left(\sqrt g\,g^{ij}\partial_j\right).$$

In spherical coordinates this contains radial and angular terms, while the inner product uses $r^2\sin\theta\,dr\,d\theta\,d\phi$. A mere coordinate change must be distinguished from physically constraining motion to a lower-dimensional surface, which changes the system itself.

When the kinetic term is restricted to a surface or a constrained coordinate, the Hamiltonian changes again. A rigid rotor, for example, has kinetic energy

$$\hat H = \frac{\hat L^2}{2I},$$

because the radial degree of freedom has been removed and only angular motion remains.

Potential Energy As Multiplication

For an ordinary conservative scalar potential, $V$ is real and the potential-energy operator acts by multiplication:

$$(\hat V\psi)(x)=V(x)\psi(x).$$

If $V$ is unbounded, this multiplication operator has its own domain: one requires $V\psi$ to belong to the Hilbert space. A complex potential instead describes an effective source or sink and is not a self-adjoint closed-system Hamiltonian.

Typical one-dimensional examples include:

• a square well, where $V(x)$ is lower in a finite region;
• a step, where $V(x)$ changes between two asymptotic values;
• a rectangular barrier, where $V(x)$ is higher in a finite region;
• a harmonic oscillator, where $V(x)=m\omega^2x^2/2$.

If $V(x,t)$ depends explicitly on time, the Hamiltonian is time dependent. Then stationary-state expansions no longer solve the problem by simply attaching phases $e^{-iE_nt/\hbar}$; noncommuting Hamiltonians at different times require the general time-dependent evolution framework.

Piecewise Potentials

Piecewise potentials are solved region by region. For example, a rectangular barrier uses one expression for $V(x)$ on the left, another inside the barrier, and another on the right. In each region, the stationary equation has simple local solutions, but the global wavefunction is fixed by matching conditions at the interfaces.

For a finite jump in $V(x)$ with constant mass, the standard kinetic term, and no singular gauge term, both $\psi$ and $\psi'$ are continuous. An infinite wall instead imposes a domain boundary, while a delta-function potential gives a controlled derivative jump. Position-dependent-mass Hamiltonians generally match the derivative combination that carries the physical current, not plain $\psi'$. These conditions are part of the Hamiltonian domain.

The compact workflow is:

$$\text{regions} \longrightarrow \text{local solutions} \longrightarrow \text{matching conditions} \longrightarrow \text{spectrum or scattering data}.$$

Use Boundary Conditions for the matching rules. Transfer matrices are a later application of the same domain and interface data.

Singular Potentials

Singular idealizations, such as

$$V(x)=\lambda\delta(x),$$

cannot be treated as ordinary finite functions. The stationary equation

$$-\frac{\hbar^2}{2m}\psi''(x) +\lambda\delta(x)\psi(x) =E\psi(x)$$

implies that $\psi$ is continuous while $\psi'$ jumps:

$$\psi'(0^+)-\psi'(0^-) = \frac{2m\lambda}{\hbar^2}\psi(0).$$

The jump condition is the physical content of this idealized singular interaction. It follows by integrating the equation distributionally across a shrinking interval. More singular interactions require a domain or self-adjoint-extension analysis rather than informal multiplication by a distribution.

Multidimensional Hamiltonians

For a particle in three-dimensional Cartesian coordinates with scalar potential $V(\mathbf r)$,

$$\hat H = -\frac{\hbar^2}{2m}\nabla^2+V(\mathbf r).$$

If $V(\mathbf r)=V(r)$ is central, spherical coordinates separate angular and radial structure. For $\psi(\mathbf r)=R_\ell(r)Y_\ell^m(\theta,\phi)$, the radial kinetic operator contains $r^{-2}\partial_r(r^2\partial_r)$. For the reduced function $u_\ell(r)=rR_\ell(r)$, the first-derivative structure is removed. In either form the separated equation contains the centrifugal term

$$\frac{\hbar^2\ell(\ell+1)}{2mr^2}.$$

This term is not an added physical potential; it comes from angular kinetic energy after separation of variables. The measure for $R_\ell$ is $r^2dr$, whereas the measure for $u_\ell$ is $dr$.

In separable Cartesian models, such as the three-dimensional box, the Hamiltonian may split into independent one-dimensional pieces:

$$\hat H = \hat H_x+\hat H_y+\hat H_z.$$

Then product wavefunctions and additive energies follow from separation of variables, provided the boundary conditions also separate.

Effective Hamiltonians

An effective Hamiltonian describes selected degrees of freedom after approximations, constraints, or reductions have been made. Common examples include:

• replacing a two-body Coulomb problem by a one-body problem with reduced mass $\mu$;
• using a rigid-rotor Hamiltonian after fixing the bond length of a diatomic molecule;
• expanding a smooth potential near a stable equilibrium to obtain a harmonic oscillator;
• using a two-level Hamiltonian after projecting onto a low-energy subspace;
• using a Born–Oppenheimer potential-energy surface after separating slow nuclear and fast electronic motion.

Effective Hamiltonians are powerful, but their assumptions must be stated. A reduced Hamiltonian is not usually valid outside the scale range, symmetry sector, or subspace used to derive it.

Electromagnetic Fields

For a particle of charge $q$ in prescribed electromagnetic potentials, the scalar-potential Hamiltonian is replaced by minimal coupling:

$$\hat H = \frac{1}{2m} \left( -i\hbar\nabla-q\mathbf A \right)^2 +q\Phi.$$

The canonical momentum $-i\hbar\nabla$ and kinetic momentum $-i\hbar\nabla-q\mathbf A$ are different. Gauge choices can change the appearance of the wavefunction and the operator, while gauge-invariant observables remain unchanged.

Use Minimal Coupling in Wave Mechanics for the detailed construction and Landau Levels for the uniform-magnetic-field model.

Domains And Boundary Conditions

The expression

$$-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$$

can describe several different systems:

System	Domain And Boundary Data	Spectrum Type
free particle on the line	$L^2(\mathbb R)$, for example $D(H)=H^2(\mathbb R)$	continuous
infinite well	$0<x<L$, $\psi(0)=\psi(L)=0$	discrete
particle on a ring	$\psi(L)=e^{i\theta}\psi(0)$ and $\psi'(L)=e^{i\theta}\psi'(0)$	discrete momenta shifted by $\theta/L$
half-line problem	$L^2(0,\infty)$ with a real Robin condition such as $\psi'(0)=\gamma\psi(0)$	boundary-condition dependent

The differential expression is the same in all four rows. The quantum problem is different because the Hilbert space and operator domain are different. Continuum normalization describes generalized spectral eigenfunctions; it is not an operator domain.

For the standard constant-mass expression on $[a,b]$, integration by parts gives the boundary form

$$\langle\phi,H\psi\rangle -\langle H\phi,\psi\rangle = -\frac{\hbar^2}{2m} \left[ \phi^*\psi'-\phi'^*\psi \right]_a^b.$$

Vanishing of this form on a proposed domain establishes symmetry. A self-adjoint Hamiltonian additionally requires the domain to equal the domain of its adjoint; zero endpoint flux alone does not prove that maximality.

This is the practical version of the symmetric-versus-self-adjoint warning. A formally symmetric differential expression may fail to generate valid unitary evolution until its domain has been specified. Self-Adjoint Operators develops the general criterion.

Construction Checklist

When building a coordinate-space Hamiltonian, record:

1. the degrees of freedom and coordinates;
2. the Hilbert space and inner product;
3. the kinetic-energy operator in those coordinates;
4. scalar potentials, constraints, and approximations;
5. singular terms or matching rules, if present;
6. electromagnetic potentials and gauge choice, if present;
7. the coordinate domain and boundary conditions;
8. the natural length, energy, or frequency scales;
9. the intended use: bound-state spectrum, scattering, time evolution, or expectation values.

This checklist prevents a compact formula from hiding physical assumptions.

Common Mistakes

• Treating $-\hbar^2d^2/(2m\,dx^2)+V(x)$ as a complete problem before specifying the domain.
• Reusing infinite-wall boundary conditions for finite walls.
• Expanding a minimally coupled Hamiltonian as if derivatives did not act on $\mathbf A$.
• Forgetting reduced mass in two-body Hamiltonians.
• Calling an effective Hamiltonian exact after projecting out degrees of freedom.
• Dropping the coordinate measure when changing variables.
• Treating singular potentials as ordinary finite jumps.

Where this is used

• Time-Independent Schrödinger Equation turns time-independent Hamiltonians into eigenvalue problems.
• Boundary Conditions explains how domains complete the operator.
• Time-Dependent Schrödinger Equation in Coordinate Space uses the Hamiltonian as the PDE generator.
• Minimal Coupling in Wave Mechanics develops the electromagnetic extension.
• Landau Levels is a model in which the vector potential, domain, and degeneracy all matter.

Exercises

1. Explain why $\hat H=-\hbar^2d^2/(2m\,dx^2)$ does not uniquely define a quantum system.

Solution

The differential expression must be paired with a Hilbert space, coordinate domain, and boundary conditions. On the full line it describes a free particle with continuous spectrum. On $0<x<L$ with $\psi(0)=\psi(L)=0$, it describes an infinite square well with discrete spectrum. On a ring it uses periodic boundary conditions and gives quantized momenta. The local expression is the same, but the domains are different.

2. A constant-mass Hamiltonian has the standard kinetic term and a finite scalar-potential jump at $x=a$, with no singular gauge term. Which matching conditions apply?

Solution

Under the stated assumptions, both the wavefunction and its first derivative are continuous:

$$\psi(a^-)=\psi(a^+), \qquad \psi'(a^-)=\psi'(a^+).$$

Derivative discontinuities arise for singular potentials such as delta functions, not for ordinary finite jumps. Position-dependent-mass or other nonstandard kinetic operators require their own current-preserving interface rule.

3. Derive the boundary form for $H=-\hbar^2d^2/(2m\,dx^2)+V(x)$ on $[a,b]$ with real $V$.

Solution

The potential terms cancel in $\langle\phi,H\psi\rangle-\langle H\phi,\psi\rangle$. Integrating the two kinetic terms by parts gives

$$\langle\phi,H\psi\rangle-\langle H\phi,\psi\rangle = -\frac{\hbar^2}{2m} \left[\phi^*\psi'-\phi'^*\psi\right]_a^b.$$

Dirichlet endpoints, real Robin endpoints, and periodic or common-phase twisted matching all make this form vanish when imposed consistently on both test functions. Vanishing proves symmetry on that domain; self-adjointness still requires equality with the adjoint domain.

4. In one dimension, expand $(-i\hbar\partial_x-qA(x))^2\psi$ without assuming that $A$ is constant.

Solution

Applying the first-order operator twice gives

$$\begin{aligned} (-i\hbar\partial_x-qA)^2\psi ={}&-\hbar^2\psi'' +iq\hbar A'\psi\\ &+2iq\hbar A\psi' +q^2A^2\psi. \end{aligned}$$

The $A'\psi$ term is missed if one treats the derivative as though it did not act on the vector potential.

5. Near a stable equilibrium $x_0$, let

$$V(x)\approx V(x_0)+\frac12V''(x_0)(x-x_0)^2.$$

Identify the effective oscillator frequency for a particle of mass $m$.

Solution

Compare the quadratic term with $m\omega^2(x-x_0)^2/2$. Thus

$$m\omega^2=V''(x_0), \qquad \omega=\sqrt{\frac{V''(x_0)}{m}}.$$

The constant $V(x_0)$ shifts all energies by the same amount and does not change the oscillator wavefunctions.

References

• G. Bonneau, J. Faraut, and G. Valent, “Self-adjoint extensions of operators and the teaching of quantum mechanics,” American Journal of Physics 69, 322–331 (2001).
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness, Academic Press, 1975.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.