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Hamiltonian (quantum mechanics)

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Quantum operator for the sum of energies of a system

Inquantum mechanics, theHamiltonian of a system is anoperator corresponding to the total energy of that system, including bothkinetic energy andpotential energy. Itsspectrum, the system'senergy spectrum or its set ofenergy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum andtime-evolution of a system, it is of fundamental importance in mostformulations of quantum theory.

The Hamiltonian is named afterWilliam Rowan Hamilton, who developed a revolutionary reformulation ofNewtonian mechanics, known asHamiltonian mechanics, which was historically important to the development of quantum physics. Similar tovector notation, it is typically denoted byH^{\displaystyle {\hat {H}}}, where the hat indicates that it is an operator. It can also be written asH{\displaystyle H} orHˇ{\displaystyle {\check {H}}}.

Introduction

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Main article:Operator (physics) § Operators in quantum mechanics

The Hamiltonian of a system represents the totalenergy of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. The Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several particles in the system, interaction between particles, kind of potential energy, time varying potential or time independent one.

Schrödinger Hamiltonian

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One particle

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By analogy withclassical mechanics, the Hamiltonian is commonly expressed as the sum ofoperators corresponding to thekinetic andpotential energies of a system in the formH^=T^+V^,{\displaystyle {\hat {H}}={\hat {T}}+{\hat {V}},}whereV^=V=V(r,t),{\displaystyle {\hat {V}}=V=V(\mathbf {r} ,t),}is thepotential energy operator andT^=p^⋅p^2m=p^22m=−ℏ22m∇2,{\displaystyle {\hat {T}}={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}={\frac {{\hat {p}}^{2}}{2m}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2},}is thekinetic energy operator in whichm{\displaystyle m} is themass of the particle, the dot denotes thedot product of vectors, andp^=−iℏ∇,{\displaystyle {\hat {p}}=-i\hbar \nabla ,}is themomentum operator where a∇{\displaystyle \nabla } is thedeloperator. Thedot product of∇{\displaystyle \nabla } with itself is theLaplacian∇2{\displaystyle \nabla ^{2}}. In three dimensions usingCartesian coordinates the Laplace operator is∇2=∂2∂x2+∂2∂y2+∂2∂z2{\displaystyle \nabla ^{2}={\frac {\partial ^{2}}{{\partial x}^{2}}}+{\frac {\partial ^{2}}{{\partial y}^{2}}}+{\frac {\partial ^{2}}{{\partial z}^{2}}}}

Although this is not the technical definition of theHamiltonian in classical mechanics, it is the form it most commonly takes. Combining these yields the form used in theSchrödinger equation:H^=T^+V^=p^⋅p^2m+V(r,t)=−ℏ22m∇2+V(r,t){\displaystyle {\begin{aligned}{\hat {H}}&={\hat {T}}+{\hat {V}}\\[6pt]&={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}+V(\mathbf {r} ,t)\\[6pt]&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\end{aligned}}}which allows one to apply the Hamiltonian to systems described by awave functionΨ(r,t){\displaystyle \Psi (\mathbf {r} ,t)}. This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics.

One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields.

Expectation value

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It can be shown that the expectation value of the Hamiltonian which gives the energy expectation value will always be greater than or equal to the minimum potential of the system.

Consider computing the expectation value of kinetic energy:T=−ℏ22m∫−∞+∞ψ∗d2ψdx2dx=−ℏ22m([ψ′(x)ψ∗(x)]−∞+∞−∫−∞+∞dψdxdψ∗dxdx)=ℏ22m∫−∞+∞|dψdx|2dx≥0{\displaystyle {\begin{aligned}T&=-{\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{+\infty }\psi ^{*}{\frac {d^{2}\psi }{dx^{2}}}\,dx\\[1ex]&=-{\frac {\hbar ^{2}}{2m}}\left({\left[\psi '(x)\psi ^{*}(x)\right]}_{-\infty }^{+\infty }-\int _{-\infty }^{+\infty }{\frac {d\psi }{dx}}{\frac {d\psi ^{*}}{dx}}\,dx\right)\\[1ex]&={\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{+\infty }\left|{\frac {d\psi }{dx}}\right|^{2}\,dx\geq 0\end{aligned}}}

Hence the expectation value of kinetic energy is always non-negative. This result can be used to calculate the expectation value of the total energy which is given for a normalized wavefunction as:E=T+⟨V(x)⟩=T+∫−∞+∞V(x)|ψ(x)|2dx≥Vmin(x)∫−∞+∞|ψ(x)|2dx≥Vmin(x){\displaystyle E=T+\langle V(x)\rangle =T+\int _{-\infty }^{+\infty }V(x)|\psi (x)|^{2}\,dx\geq V_{\text{min}}(x)\int _{-\infty }^{+\infty }|\psi (x)|^{2}\,dx\geq V_{\text{min}}(x)}which complete the proof. Similarly, the condition can be generalized to any higher dimensions usingdivergence theorem.

Many particles

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The formalism can be extended toN{\displaystyle N} particles:H^=∑n=1NT^n+V^{\displaystyle {\hat {H}}=\sum _{n=1}^{N}{\hat {T}}_{n}+{\hat {V}}}whereV^=V(r1,r2,…,rN,t),{\displaystyle {\hat {V}}=V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t),}is the potential energy function, now a function of the spatial configuration of the system and time (a particular set of spatial positions at some instant of time defines a configuration) andT^n=p^n⋅p^n2mn=−ℏ22mn∇n2{\displaystyle {\hat {T}}_{n}={\frac {\mathbf {\hat {p}} _{n}\cdot \mathbf {\hat {p}} _{n}}{2m_{n}}}=-{\frac {\hbar ^{2}}{2m_{n}}}\nabla _{n}^{2}}is the kinetic energy operator of particlen{\displaystyle n},∇n{\displaystyle \nabla _{n}} is the gradient for particlen{\displaystyle n}, and∇n2{\displaystyle \nabla _{n}^{2}} is the Laplacian for particlen:∇n2=∂2∂xn2+∂2∂yn2+∂2∂zn2,{\displaystyle \nabla _{n}^{2}={\frac {\partial ^{2}}{\partial x_{n}^{2}}}+{\frac {\partial ^{2}}{\partial y_{n}^{2}}}+{\frac {\partial ^{2}}{\partial z_{n}^{2}}},}

Combining these yields the Schrödinger Hamiltonian for theN{\displaystyle N}-particle case:H^=∑n=1NT^n+V^=∑n=1Np^n⋅p^n2mn+V(r1,r2,…,rN,t)=−ℏ22∑n=1N1mn∇n2+V(r1,r2,…,rN,t){\displaystyle {\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\hat {T}}_{n}+{\hat {V}}\\[6pt]&=\sum _{n=1}^{N}{\frac {\mathbf {\hat {p}} _{n}\cdot \mathbf {\hat {p}} _{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)\\[6pt]&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)\end{aligned}}}

However, complications can arise in themany-body problem. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles:−ℏ22M∇i⋅∇j{\displaystyle -{\frac {\hbar ^{2}}{2M}}\nabla _{i}\cdot \nabla _{j}}whereM{\displaystyle M} denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known asmass polarization terms, and appear in the Hamiltonian of many-electron atoms (see below).

ForN{\displaystyle N} interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy functionV{\displaystyle V} isnot simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle.

For non-interacting particles, i.e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle,[1] that isV=∑i=1NV(ri,t)=V(r1,t)+V(r2,t)+⋯+V(rN,t){\displaystyle V=\sum _{i=1}^{N}V(\mathbf {r} _{i},t)=V(\mathbf {r} _{1},t)+V(\mathbf {r} _{2},t)+\cdots +V(\mathbf {r} _{N},t)}

The general form of the Hamiltonian in this case is:H^=−ℏ22∑i=1N1mi∇i2+∑i=1NVi=∑i=1N(−ℏ22mi∇i2+Vi)=∑i=1NH^i{\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}{\frac {1}{m_{i}}}\nabla _{i}^{2}+\sum _{i=1}^{N}V_{i}\\[6pt]&=\sum _{i=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}+V_{i}\right)\\[6pt]&=\sum _{i=1}^{N}{\hat {H}}_{i}\end{aligned}}}where the sum is taken over all particles and their corresponding potentials; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle. This is an idealized situation—in practice the particles are almost always influenced by some potential, and there are many-body interactions. One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they interact with each other by Coulomb interaction (electrostatic force), as shown below.

Schrödinger equation

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Main article:Schrödinger equation

The Hamiltonian generates the time evolution of quantum states. If|ψ(t)⟩{\displaystyle \left|\psi (t)\right\rangle } is the state of the system at timet{\displaystyle t}, thenH|ψ(t)⟩=iℏd dt|ψ(t)⟩.{\displaystyle H\left|\psi (t)\right\rangle =i\hbar {d \over \ dt}\left|\psi (t)\right\rangle .}

This equation is theSchrödinger equation. It takes the same form as theHamilton–Jacobi equation, which is one of the reasonsH{\displaystyle H} is also called the Hamiltonian. Given the state at some initial time (t=0{\displaystyle t=0}), we can solve it to obtain the state at any subsequent time. In particular, ifH{\displaystyle H} is independent of time, then|ψ(t)⟩=e−iHt/ℏ|ψ(0)⟩.{\displaystyle \left|\psi (t)\right\rangle =e^{-iHt/\hbar }\left|\psi (0)\right\rangle .}

Theexponential operator on the right hand side of the Schrödinger equation is usually defined by the correspondingpower series inH{\displaystyle H}. One might notice that taking polynomials or power series ofunbounded operators that are not defined everywhere may not make mathematical sense. Rigorously, to take functions of unbounded operators, afunctional calculus is required. In the case of the exponential function, thecontinuous, or just theholomorphic functional calculus suffices. We note again, however, that for common calculations the physicists' formulation is quite sufficient.

By the *-homomorphism property of the functional calculus, the operatorU=e−iHt/ℏ{\displaystyle U=e^{-iHt/\hbar }}is aunitary operator. It is thetime evolution operator orpropagator of a closed quantum system. If the Hamiltonian is time-independent,{U(t)}{\displaystyle \{U(t)\}} form aone parameter unitary group (more than asemigroup); this gives rise to the physical principle ofdetailed balance.

Dirac formalism

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However, in themore general formalism ofDirac, the Hamiltonian is typically implemented as an operator on aHilbert space in the following way:

Theeigenkets ofH{\displaystyle H}, denoted|a⟩{\displaystyle \left|a\right\rangle }, provide anorthonormal basis for the Hilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted{Ea}{\displaystyle \{E_{a}\}}, solving the equation:H|a⟩=Ea|a⟩.{\displaystyle H\left|a\right\rangle =E_{a}\left|a\right\rangle .}

SinceH{\displaystyle H} is aHermitian operator, the energy is always areal number.

From a mathematically rigorous point of view, care must be taken with the above assumptions. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with thespectrum of an operator). However, all routine quantum mechanical calculations can be done using the physical formulation.[clarification needed]

Expressions for the Hamiltonian

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Following are expressions for the Hamiltonian in a number of situations.[2] Typical ways to classify the expressions are the number of particles, number of dimensions, and the nature of the potential energy function—importantly space and time dependence. Masses are denoted bym{\displaystyle m}, and charges byq{\displaystyle q}.

Free particle

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The particle is not bound by any potential energy, so the potential is zero and this Hamiltonian is the simplest. For one dimension:H^=−ℏ22m∂2∂x2{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}}and in higher dimensions:H^=−ℏ22m∇2{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}}

Constant-potential well

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For a particle in a region of constant potentialV=V0{\displaystyle V=V_{0}} (no dependence on space or time), in one dimension, the Hamiltonian is:H^=−ℏ22m∂2∂x2+V0{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V_{0}}in three dimensionsH^=−ℏ22m∇2+V0{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{0}}

This applies to the elementary "particle in a box" problem, andstep potentials.

Simple harmonic oscillator

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For asimple harmonic oscillator in one dimension, the potential varies with position (but not time), according to:V=k2x2=mω22x2{\displaystyle V={\frac {k}{2}}x^{2}={\frac {m\omega ^{2}}{2}}x^{2}}where theangular frequencyω{\displaystyle \omega }, effectivespring constantk{\displaystyle k}, and massm{\displaystyle m} of the oscillator satisfy:ω2=km{\displaystyle \omega ^{2}={\frac {k}{m}}}so the Hamiltonian is:H^=−ℏ22m∂2∂x2+mω22x2{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}}

For three dimensions, this becomesH^=−ℏ22m∇2+mω22r2{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\frac {m\omega ^{2}}{2}}r^{2}}where the three-dimensional position vectorr{\displaystyle \mathbf {r} } using Cartesian coordinates is(x,y,z){\displaystyle (x,y,z)}, its magnitude isr2=r⋅r=|r|2=x2+y2+z2{\displaystyle r^{2}=\mathbf {r} \cdot \mathbf {r} =|\mathbf {r} |^{2}=x^{2}+y^{2}+z^{2}}

Writing the Hamiltonian out in full shows it is simply the sum of the one-dimensional Hamiltonians in each direction:H^=−ℏ22m(∂2∂x2+∂2∂y2+∂2∂z2)+mω22(x2+y2+z2)=(−ℏ22m∂2∂x2+mω22x2)+(−ℏ22m∂2∂y2+mω22y2)+(−ℏ22m∂2∂z2+mω22z2){\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)+{\frac {m\omega ^{2}}{2}}\left(x^{2}+y^{2}+z^{2}\right)\\[6pt]&=\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}\right)+\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {m\omega ^{2}}{2}}y^{2}\right)+\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}+{\frac {m\omega ^{2}}{2}}z^{2}\right)\end{aligned}}}

Rigid rotor

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For arigid rotor—i.e., system of particles which can rotate freely about any axes, not bound in any potential (such as free molecules with negligible vibrationaldegrees of freedom, say due todouble ortriplechemical bonds), the Hamiltonian is:H^=−ℏ22IxxJ^x2−ℏ22IyyJ^y2−ℏ22IzzJ^z2{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2I_{xx}}}{\hat {J}}_{x}^{2}-{\frac {\hbar ^{2}}{2I_{yy}}}{\hat {J}}_{y}^{2}-{\frac {\hbar ^{2}}{2I_{zz}}}{\hat {J}}_{z}^{2}}whereIxx{\displaystyle I_{xx}},Iyy{\displaystyle I_{yy}}, andIzz{\displaystyle I_{zz}} are themoment of inertia components (technically the diagonal elements of themoment of inertia tensor), andJ^x{\displaystyle {\hat {J}}_{x}},J^y{\displaystyle {\hat {J}}_{y}}, andJ^z{\displaystyle {\hat {J}}_{z}} are the totalangular momentum operators (components), about thex{\displaystyle x},y{\displaystyle y}, andz{\displaystyle z} axes respectively.

Electrostatic (Coulomb) potential

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TheCoulomb potential energy for two point chargesq1{\displaystyle q_{1}} andq2{\displaystyle q_{2}} (i.e., those that have no spatial extent independently), in three dimensions, is (inSI units—rather thanGaussian units which are frequently used inelectromagnetism):V=q1q24πε0|r|{\displaystyle V={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}|\mathbf {r} |}}}

However, this is only the potential for one point charge due to another. If there are many charged particles, each charge has a potential energy due to every other point charge (except itself). ForN{\displaystyle N} charges, the potential energy of chargeqj{\displaystyle q_{j}} due to all other charges is (see alsoElectrostatic potential energy stored in a configuration of discrete point charges):[3]Vj=12∑i≠jqiϕ(ri)=18πε0∑i≠jqiqj|ri−rj|{\displaystyle V_{j}={\frac {1}{2}}\sum _{i\neq j}q_{i}\phi (\mathbf {r} _{i})={\frac {1}{8\pi \varepsilon _{0}}}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}whereϕ(ri){\displaystyle \phi (\mathbf {r} _{i})} is the electrostatic potential of chargeqj{\displaystyle q_{j}} atri{\displaystyle \mathbf {r} _{i}}. The total potential of the system is then the sum overj{\displaystyle j}:V=18πε0∑j=1N∑i≠jqiqj|ri−rj|{\displaystyle V={\frac {1}{8\pi \varepsilon _{0}}}\sum _{j=1}^{N}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}so the Hamiltonian is:H^=−ℏ22∑j=1N1mj∇j2+18πε0∑j=1N∑i≠jqiqj|ri−rj|=∑j=1N(−ℏ22mj∇j2+18πε0∑i≠jqiqj|ri−rj|){\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2}}\sum _{j=1}^{N}{\frac {1}{m_{j}}}\nabla _{j}^{2}+{\frac {1}{8\pi \varepsilon _{0}}}\sum _{j=1}^{N}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\\&=\sum _{j=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{j}}}\nabla _{j}^{2}+{\frac {1}{8\pi \varepsilon _{0}}}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\right)\\\end{aligned}}}

Electric dipole in an electric field

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For anelectric dipole momentd{\displaystyle \mathbf {d} } constituting charges of magnitudeq{\displaystyle q}, in a uniform,electrostatic field (time-independent)E{\displaystyle \mathbf {E} }, positioned in one place, the potential is:V=−d^⋅E{\displaystyle V=-\mathbf {\hat {d}} \cdot \mathbf {E} }the dipole moment itself is the operatord^=qr^{\displaystyle \mathbf {\hat {d}} =q\mathbf {\hat {r}} }

Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy:H^=−d^⋅E=−qr^⋅E{\displaystyle {\hat {H}}=-\mathbf {\hat {d}} \cdot \mathbf {E} =-q\mathbf {\hat {r}} \cdot \mathbf {E} }

Magnetic dipole in a magnetic field

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For a magnetic dipole momentμ{\displaystyle {\boldsymbol {\mu }}} in a uniform, magnetostatic field (time-independent)B{\displaystyle \mathbf {B} }, positioned in one place, the potential is:V=−μ⋅B{\displaystyle V=-{\boldsymbol {\mu }}\cdot \mathbf {B} }

Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy:H^=−μ⋅B{\displaystyle {\hat {H}}=-{\boldsymbol {\mu }}\cdot \mathbf {B} }

For aspin-1⁄2 particle, the corresponding spin magnetic moment is:[4]μS=gse2mS{\displaystyle {\boldsymbol {\mu }}_{S}={\frac {g_{s}e}{2m}}\mathbf {S} }wheregs{\displaystyle g_{s}} is the "sping-factor" (not to be confused with thegyromagnetic ratio),e{\displaystyle e} is the electron charge,S{\displaystyle \mathbf {S} } is thespin operator vector, whose components are thePauli matrices, henceH^=gse2mS⋅B{\displaystyle {\hat {H}}={\frac {g_{s}e}{2m}}\mathbf {S} \cdot \mathbf {B} }

Charged particle in an electromagnetic field

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For a particle with massm{\displaystyle m} and chargeq{\displaystyle q} in an electromagnetic field, described by thescalar potentialϕ{\displaystyle \phi } andvector potentialA{\displaystyle \mathbf {A} }, there are two parts to the Hamiltonian to substitute for.[1] The canonical momentum operatorp^{\displaystyle \mathbf {\hat {p}} }, which includes a contribution from theA{\displaystyle \mathbf {A} } field and fulfils thecanonical commutation relation, must be quantized;p^=mr˙+qA,{\displaystyle \mathbf {\hat {p}} =m{\dot {\mathbf {r} }}+q\mathbf {A} ,}wheremr˙{\displaystyle m{\dot {\mathbf {r} }}} is thekinetic momentum. The quantization prescription readsp^=−iℏ∇,{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla ,}so the corresponding kinetic energy operator isT^=12mr˙⋅r˙=12m(p^−qA)2{\displaystyle {\hat {T}}={\frac {1}{2}}m{\dot {\mathbf {r} }}\cdot {\dot {\mathbf {r} }}={\frac {1}{2m}}\left(\mathbf {\hat {p}} -q\mathbf {A} \right)^{2}}and the potential energy, which is due to theϕ{\displaystyle \phi } field, is given byV^=qϕ.{\displaystyle {\hat {V}}=q\phi .}

Casting all of these into the Hamiltonian givesH^=12m(−iℏ∇−qA)2+qϕ.{\displaystyle {\hat {H}}={\frac {1}{2m}}\left(-i\hbar \nabla -q\mathbf {A} \right)^{2}+q\phi .}

Energy eigenket degeneracy, symmetry, and conservation laws

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In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of itswavelength. A wave propagating in thex{\displaystyle x} direction is a different state from one propagating in they{\displaystyle y} direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to bedegenerate.

It turns out thatdegeneracy occurs whenever a nontrivialunitary operatorU{\displaystyle U}commutes with the Hamiltonian. To see this, suppose that|a⟩{\displaystyle |a\rangle } is an energy eigenket. ThenU|a⟩{\displaystyle U|a\rangle } is an energy eigenket with the same eigenvalue, sinceUH|a⟩=UEa|a⟩=Ea(U|a⟩)=H(U|a⟩).{\displaystyle UH|a\rangle =UE_{a}|a\rangle =E_{a}(U|a\rangle )=H\;(U|a\rangle ).}

SinceU{\displaystyle U} is nontrivial, at least one pair of|a⟩{\displaystyle |a\rangle } andU|a⟩{\displaystyle U|a\rangle } must represent distinct states. Therefore,H{\displaystyle H} has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is therotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.

The existence of a symmetry operator implies the existence of aconserved observable. LetG{\displaystyle G} be the Hermitian generator ofU{\displaystyle U}:U=I−iεG+O(ε2){\displaystyle U=I-i\varepsilon G+O(\varepsilon ^{2})}

It is straightforward to show that ifU{\displaystyle U} commutes withH{\displaystyle H}, then so doesG{\displaystyle G}:[H,G]=0{\displaystyle [H,G]=0}

Therefore,∂∂t⟨ψ(t)|G|ψ(t)⟩=1iℏ⟨ψ(t)|[G,H]|ψ(t)⟩=0.{\displaystyle {\frac {\partial }{\partial t}}\langle \psi (t)|G|\psi (t)\rangle ={\frac {1}{i\hbar }}\langle \psi (t)|[G,H]|\psi (t)\rangle =0.}

In obtaining this result, we have used the Schrödinger equation, as well as itsdual,⟨ψ(t)|H=−iℏd dt⟨ψ(t)|.{\displaystyle \langle \psi (t)|H=-i\hbar {d \over \ dt}\langle \psi (t)|.}

Thus, theexpected value of the observableG{\displaystyle G} is conserved for any state of the system. In the case of the free particle, the conserved quantity is theangular momentum.

Hamilton's equations

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Hamilton's equations in classicalHamiltonian mechanics have a direct analogy in quantum mechanics. Suppose we have a set of basis states{|n⟩}{\displaystyle \left\{\left|n\right\rangle \right\}}, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,⟨n′|n⟩=δnn′{\displaystyle \langle n'|n\rangle =\delta _{nn'}}

Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time.

The instantaneous state of the system at timet{\displaystyle t},|ψ(t)⟩{\displaystyle \left|\psi \left(t\right)\right\rangle }, can be expanded in terms of these basis states:|ψ(t)⟩=∑nan(t)|n⟩{\displaystyle |\psi (t)\rangle =\sum _{n}a_{n}(t)|n\rangle }wherean(t)=⟨n|ψ(t)⟩.{\displaystyle a_{n}(t)=\langle n|\psi (t)\rangle .}

The coefficientsan(t){\displaystyle a_{n}(t)} arecomplex variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole.

The expectation value of the Hamiltonian of this state, which is also the mean energy, is⟨H(t)⟩=def⟨ψ(t)|H|ψ(t)⟩=∑nn′an′∗an⟨n′|H|n⟩{\displaystyle \langle H(t)\rangle \mathrel {\stackrel {\mathrm {def} }{=}} \langle \psi (t)|H|\psi (t)\rangle =\sum _{nn'}a_{n'}^{*}a_{n}\langle n'|H|n\rangle }where the last step was obtained by expanding|ψ(t)⟩{\displaystyle \left|\psi \left(t\right)\right\rangle } in terms of the basis states.

Eachan(t){\displaystyle a_{n}(t)} actually corresponds totwo independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we usean(t){\displaystyle a_{n}(t)} and itscomplex conjugatean∗(t){\displaystyle a_{n}^{*}(t)}. With this choice of independent variables, we can calculate thepartial derivative∂⟨H⟩∂an′∗=∑nan⟨n′|H|n⟩=⟨n′|H|ψ⟩{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n'}^{*}}}=\sum _{n}a_{n}\langle n'|H|n\rangle =\langle n'|H|\psi \rangle }

By applying theSchrödinger equation and using the orthonormality of the basis states, this further reduces to∂⟨H⟩∂an′∗=iℏ∂an′∂t{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n'}^{*}}}=i\hbar {\frac {\partial a_{n'}}{\partial t}}}

Similarly, one can show that∂⟨H⟩∂an=−iℏ∂an∗∂t{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n}}}=-i\hbar {\frac {\partial a_{n}^{*}}{\partial t}}}

If we define "conjugate momentum" variablesπn{\displaystyle \pi _{n}} byπn(t)=iℏan∗(t){\displaystyle \pi _{n}(t)=i\hbar a_{n}^{*}(t)}then the above equations become∂⟨H⟩∂πn=∂an∂t,∂⟨H⟩∂an=−∂πn∂t{\displaystyle {\frac {\partial \langle H\rangle }{\partial \pi _{n}}}={\frac {\partial a_{n}}{\partial t}},\quad {\frac {\partial \langle H\rangle }{\partial a_{n}}}=-{\frac {\partial \pi _{n}}{\partial t}}}which is precisely the form of Hamilton's equations, with thean{\displaystyle a_{n}}s as the generalized coordinates, theπn{\displaystyle \pi _{n}}s as the conjugate momenta, and⟨H⟩{\displaystyle \langle H\rangle } taking the place of the classical Hamiltonian.

See also

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References

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  1. ^abResnick, R.; Eisberg, R. (1985).Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd ed.). John Wiley & Sons.ISBN 0-471-87373-X.
  2. ^Atkins, P. W. (1974).Quanta: A Handbook of Concepts. Oxford University Press.ISBN 0-19-855493-1.
  3. ^Grant, I. S.; Phillips, W. R. (2008).Electromagnetism. Manchester Physics Series (2nd ed.).ISBN 978-0-471-92712-9.
  4. ^Bransden, B. H.; Joachain, C. J. (1983).Physics of Atoms and Molecules. Longman.ISBN 0-582-44401-2.

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