Lecture 1

In this course we will develop some of the mathematics needed to describe the physical theory of Quantum Mechanics. Quantum Mechanics was developed in the 1920s by Bohr, Schroedinger, Heisenberg and others in order to describe atomic physics. In order to describe quantum mechanics it is useful, first, to understand the theory it extends namely Classical Mechanics. Classical Mechanics is the physics of Newton. It is the physics of the everyday world. Although Quantum Mechanics is not Classical Mechanics, many ideas are common between the two.

A PDF version of this lecture is available here.

Newtonian Mechanics

Newtons three laws of motion are the basis of classical mechanics. Taken together the first two tell us that a point-like particle {P} with position {\mathbf{x}_{P}\in\mathbb{R}^{3}} and velocity {\mathbf{v}_{P}=\frac{d\mathbf{x}_{P}}{dt}} satisfies the second order law of motion

\displaystyle  \frac{dm_{P}\mathbf{v}_{P}}{dt}=\mathbf{F}_{P} \ \ \ \ \ (1)

where {\mathbf{F}_{P}} is the total force acting on the particle, {m_{P}} is the mass of the particle and {t} is time. Usually, the mass {m_{P}} of a particle is conserved, i.e., constant in time so that {\frac{dm_{P}\mathbf{v}_{p}}{dt}=m_{P}\frac{d\mathbf{v}_{P}}{dt}}. The quantity {m_{P}\mathbf{v}_{P}} is called the momentum of the particle {P} and is denoted {\mathbf{p}_{P}}. So Newton’s law of motion (1) simply states that the rate of change of momentum is the total force {\mathbf{F}_{P}} acting on {P}. We say total force, because {\mathbf{F}_{P}} is supposed to be the sum of all forces acting on the particle, {\mathbf{F}_{P}=\sum_{j}\mathbf{F}_{P,j}} where {\mathbf{F}_{P,j}} represent the various “fundamental” forces acting on the particle {P}. Newton’s third law — “to every action there is an equal and opposite reaction” — is taken to mean that the particle {P} must exert a force {-\mathbf{F}_{P,j}} on the origin of the force {\mathbf{F}_{P,j}}.

For instance if one considers an ensemble of point-like particles {P_{j}}, {j=1,\ldots,N}, with positions {\mathbf{x}_{j}}, masses {m_{j}} and velocities {\mathbf{v}_{j}}, {j=1,\ldots,N} then we have

\displaystyle  \frac{d\mathbf{p}_{j}}{dt}=\sum_{i\neq j}\mathbf{F}_{j,i}

where {\mathbf{p}_{j}=m_{j}\mathbf{v}_{j}} is the momentum of the {j}-th particle and {\mathbf{F}_{j,i}} is the force exerted on the {j}-th particle by the {i}-th. Newton’s third law tells us that {\mathbf{F}_{j,i}+\mathbf{F}_{i,j}=0}. In particular we find that

\displaystyle  \frac{d}{dt}\sum_{j}\mathbf{p}_{j}=0,

which is the principle of conservation of (the total) momentum.

Conservative forces and Hamiltonian Mechanics

In practice the equations of motions needed to describe physical systems have a bit more structure. It turns out that many forces are conservative, meaning that the force {\mathbf{F}_{P}} depends only on the positon of the particle {P} and that

\displaystyle  \mathbf{F}_{P}(\mathbf{x}_{P})=-\nabla V_{P}(\mathbf{x}_{P}) \ \ \ \ \ (2)

where {V_{P}} is the potential energy of {P}. Note that the potential energy, when it exists, is only defined up to an additive constant. (For a force field {\mathbf{F}_{P}} defined on all of {\mathbb{R}^{3}} a necessary and sufficient condition for {V_{P}} to exist is that {\nabla\times\mathbf{F}_{P}=0} — this is the PoincarÈ Lemma. If the force field is defined on a domain with more complicated topology then this condition is necessary but may not be sufficient.)

A particle {P} in a conservative force field {\mathbf{F}_{P}} satisfies the principle of conservation of energy

\displaystyle  \frac{d}{dt}\left\{ \frac{1}{2}m_{P}\left|\mathbf{v}_{P}\right|^{2}+V_{P}(\mathbf{x}_{P})\right\} =0

as one can readily check. Here {E_{P}=\frac{1}{2}m_{P}\left|\mathbf{v}_{P}\right|^{2}+V_{P}(\mathbf{x}_{P})} is the total energy of {P} and {\frac{1}{2}m_{P}\left|\mathbf{v}_{P}\right|^{2}} is the kinetic energy. More generally, if the potential energy function {V_{P}} depends explicitly on time as well as the position {\mathbf{x}_{P}} of the particle {P}, then

\displaystyle  \frac{d}{dt}\left\{ \frac{1}{2}m_{P}\left|\mathbf{v}_{P}\right|^{2}+V_{P}(\mathbf{x}_{P},t)\right\} =\frac{\partial V_{P}}{\partial t}\left(\mathbf{x}_{P},t\right). \ \ \ \ \ (3)

(We say that {V_{P}} depends explicitly on time here because the potential energy of the particle {P} always depends on time implicitly through the motion of the particle.)

The Hamiltonian function for {P} is a function of the position and the momentum

\displaystyle  H_{P}(\mathbf{x},\mathbf{p})=\frac{1}{2m_{P}}\left|\mathbf{p}\right|^{2}+V_{P}(\mathbf{x}).

(Note that if {\mathbf{p}_{P}=m_{P}\mathbf{v}_{P}} then {H_{P}} agrees with the total energy.) The point of view of Hamiltonian mechanics is that the equations of motion for {P} may be written as follows

\displaystyle  \frac{d\mathbf{p}_{P}}{dt}=-\frac{\partial H_{P}}{\partial\mathbf{x}}(\mathbf{x}_{P},\mathbf{p}_{P}),\quad\frac{d\mathbf{x}_{P}}{dt}=\frac{\partial H_{P}}{\partial\mathbf{p}}(\mathbf{x}_{P},\mathbf{p}_{P}). \ \ \ \ \ (4)

Indeed, since {\frac{\partial H}{\partial\mathbf{x}}=\nabla V_{P}}, in view of (2) the first identity is just (1). On the other hand {\frac{\partial H}{\partial\mathbf{p}}=\frac{\mathbf{p}}{m_{P}}} so the second identity recovers {\mathbf{p}_{P}=m_{P}\mathbf{v}_{P}}. Note that the conservation of energy follows

\displaystyle  \frac{d}{dt}H_{P}\left(\mathbf{x}_{P},\mathbf{p}_{P}\right)=\frac{\partial H_{P}}{\partial\mathbf{x}}\frac{d\mathbf{x}_{P}}{dt}+\frac{\partial H_{P}}{\partial\mathbf{p}}\frac{d\mathbf{p}_{P}}{dt}=-\frac{d\mathbf{p}_{P}}{dt}\frac{d\mathbf{x}_{P}}{dt}+\frac{d\mathbf{x}_{P}}{dt}\frac{d\mathbf{p}_{P}}{dt}=0.

For a system of point particles with positions {\mathbf{x}_{j}}, momentums {\mathbf{p}_{j}} and masses {m_{j}}, {j=1,\ldots,N}, the Hamiltonian Equations of motion are

\displaystyle  \frac{d\mathbf{p}_{j}}{dt}=-\frac{\partial H}{\partial\mathbf{x}_{j}},\quad\frac{d\mathbf{x}_{j}}{dt}=\frac{\partial H}{\partial\mathbf{p}_{j}} \ \ \ \ \ (5)

where the Hamiltonian function {H} is a function of the positions and momenta. Note that as above we have

\displaystyle  \frac{d}{dt}H\left(\mathbf{x}_{1},\ldots,\mathbf{x}_{N},\mathbf{p}_{1},\ldots,\mathbf{p}_{N}\right)=0 \ \ \ \ \ (6)

along a trajectory satisfying (5). For a system of particles interacting with one another via forces {\mathbf{F}_{i,j}} as above we have

\displaystyle  H(\mathbf{x}_{1,}\ldots,\mathbf{x}_{N},\mathbf{p}_{1},\ldots,\mathbf{p}_{N})=\sum_{j=1}^{N}\frac{\left|\mathbf{p}_{j}\right|^{2}}{2m_{j}}+\sum_{i<j}V_{i,j}\left(\mathbf{x}_{i}-\mathbf{x}_{j}\right)+\sum_{i=1}^{N}V_{i}\left(\mathbf{x}_{i}\right), \ \ \ \ \ (7)

where {-\nabla V_{i,j}\left(\mathbf{x}_{i}-\mathbf{x}_{j}\right)} is the force exerted on the {i}-th particle by the {j}-th — the dependence of {H} only on the difference {\mathbf{x}_{i}-\mathbf{x}_{j}} is guaranteed by Newton’s third law. We have introduced here an additional term {\sum_{i}V_{i}} giving the interaction of the particles with an “external field” or “background potential.”

Phase space and the algebra of observables

The phase space in classical mechanics is the space {\mathcal{M}=\mathbb{R}^{3N}\times\mathbb{R}^{3N}} parameterized by the coordinates {(\mathbf{x}_{1},\ldots,\mathbf{x}_{N})} and momenta {\left(\mathbf{p}_{1},\ldots,\mathbf{p}_{N}\right)}. To simplify notation we will denote {(\mathbf{x}_{1},\ldots,\mathbf{x}_{N})} by {\mathbf{X}\in\mathbb{R}^{3N}} and {\left(\mathbf{p}_{1},\ldots,\mathbf{p}_{N}\right)} by {\mathbf{P}\in\mathbb{R}^{3N}} so that the Hamiltonian equations of motion (5) are just

\displaystyle  \frac{d\mathbf{P}}{dt}(t)=-\frac{\partial H}{\partial\mathbf{X}}\left(\mathbf{X}(t),\mathbf{P}(t)\right),\quad\frac{d\mathbf{X}}{dt}(t)=\frac{\partial H}{\partial\mathbf{P}}\left(\mathbf{X}(t),\mathbf{P}(t)\right). \ \ \ \ \ (8)

For each {s\ge0}, let {G_{s}:\mathcal{M}\rightarrow\mathcal{M}} be the evolution map for (8), that is

\displaystyle  G_{s}\left(\mathbf{X},\mathbf{P}\right)=\left(\mathbf{X}(s),\mathbf{P}(s)\right)

where {\left(\mathbf{X}(t),\mathbf{P}\left(t\right)\right)} satisfy (8) with {\mathbf{X}(0)=\mathbf{X}} and {\mathbf{P}\left(0\right)=\mathbf{P}}. This definition of {G_{s}} is quite implicit — and not necessarily easy to compute.

Note that in order to define {G_{s}} there must be a unique solution to (8) up to time {s} for any initial condition {\left(\mathbf{X}(0),\mathbf{P}\left(0\right)\right)}. A sufficient condition for this would be that the derivative {\frac{\partial H}{\partial\mathbf{X}}} and {\frac{\partial H}{\partial\mathbf{P}}} are Lipschitz functions on phase space. If {G_{s}} fails to exist the solution could breakdown in two differnt ways. Here are two examples on the phase space {\mathbb{R}^{2}}:

  • Consider the Hamiltonian {H(x,p)=p^{2}-x^{4}}. Note that the force {4x^{3}} pushes the particle away from the origin and increases in magnitude as {x} increases. In fact there are solutions that “escape to infinity in finite time.” For instance, with the initial condition {x(0)>0} and {p(0)=x(0)^{2}}, the solution is

    \displaystyle  x(t)=\frac{x(0)}{1-x(0)t},\quad t<\frac{1}{x(0)}.

  • Consider the Hamiltonian {H(x,p)=\frac{1}{2}p^{2}-\frac{1}{\left|x\right|}} defined for {x\neq0}. In this case the force {\frac{x}{\left|x\right|^{3}}} can draw the particle into the singular point {x=0} in finite time. In fact, if the initial condition has negative energy — {H\left(x(0),p(0)\right)<0} —, then {x(t)=0} at some finite time {t} beyond which the solution is not defined.

The second example is of physical relevance since it can be generalized to systems of point particles interacting according to Newton’s law of gravitation

\displaystyle  \mathbf{F}_{i,j}=-Gm_{i}m_{j}\frac{\mathbf{x}_{i}-\mathbf{x}_{j}}{\left|\mathbf{x}_{i}-\mathbf{x}_{j}\right|^{2}}

which corresponds to the potentials {V_{i,j}\left(\mathbf{x}\right)=-Gm_{i}m_{j}\frac{1}{\left|\mathbf{x}\right|}}. Gravity can (and does!) cause particles to collide in finite time. The equations of gravitation alone, however, do not describe the physics of collision.

Henceforth we will assume that the Hamiltonian is nice enough that there is a uniqe solution to (8) up to some fixed time {s} for every initial condition. Since {G_{s}} has the group property

\displaystyle  G_{s_{1}+s_{2}}=G_{s_{1}}\circ G_{s_{2}} \ \ \ \ \ (9)

it follows that there is a unique solution to (8) for all time and for every initial condition. Note that {G_{0}=Id}, the identity map. It also makes sense to define {G_{s}} for negative times {s<0}, by integrating the equations (8) “back in time.” In fact the group property (9) for positive and negative times, so {G_{s}} is an invertible map (one-to-one and onto) for each {s} and we have

\displaystyle  G_{-s}=G_{s}^{-1}

The group property still holds so, in fact, {s\mapsto G_{s}} is a one parameter group of transformations of phase space.

An observable in classical mechanics is an infinitely differentiable real valued function {f} on phase space. Think of {f} as some property of the system that can be “observed.” (We take observables to be infinitely differentiable for technical reasons.) Let {\mathcal{A}=C^{\infty}\left(\mathcal{M}\right)} denote the collection of observables. {\mathcal{A}} is an algebra, called the algebra of observables, since under the usual operations

\displaystyle  f,g\in\mathcal{A},\ a,b\in\mathbb{R}\implies af+bg,\ fg\in\mathcal{A}.

The transformations {G_{s}} induce transformations {U_{s}} of the algebra {\mathcal{A}} as follows:

\displaystyle  U_{s}f:=f\circ G_{s}.

That is

\displaystyle  U_{s}f(\mathbf{X},\mathbf{P})=f(\mathbf{X}(s),\mathbf{P}(s))

where {\left(\mathbf{X}(t),\mathbf{P}(t)\right)} satisfy (8) with {\mathbf{X}(0)=\mathbf{X}} and {\mathbf{P}(0)=\mathbf{P}}. Thus,

\displaystyle  \frac{d}{ds}U_{s}f(\mathbf{X},\mathbf{P})=\frac{\partial f}{\partial\mathbf{X}}\left(\mathbf{X}(s),\mathbf{P}(s)\right)\cdot\frac{\partial H}{\partial\mathbf{P}}\left(\mathbf{X}(s),\mathbf{P}(s)\right)-\frac{\partial f}{\partial\mathbf{P}}\left(\mathbf{X}(s),\mathbf{P}(s)\right)\cdot\frac{\partial H}{\partial\mathbf{X}}\left(\mathbf{X}(s),\mathbf{P}(s)\right).

Applying the group property {U_{s+t}=U_{s}\circ U_{t}} we obtain

\displaystyle  \frac{d}{ds}U_{s+t}f(\mathbf{X},\mathbf{P})=\left[\frac{\partial U_{t}f}{\partial\mathbf{X}}\cdot\frac{\partial H}{\partial\mathbf{P}}-\frac{\partial U_{t}f}{\partial\mathbf{P}}\cdot\frac{\partial H}{\partial\mathbf{X}}\right]\left(\mathbf{X}(s),\mathbf{P}(s)\right)

where we have noted that {U_{t}H=H} by conservation of energy. Finally taking {s\downarrow0} we obtain

\displaystyle  \frac{d}{dt}U_{t}f=\left\{ H,U_{t}f\right\}

where the Poisson bracket {\left\{ f,g\right\} } of two observables is defined to be

\displaystyle  \left\{ f,g\right\} =\frac{\partial f}{\partial\mathbf{P}}\cdot\frac{\partial g}{\partial\mathbf{X}}-\frac{\partial f}{\partial\mathbf{X}}\cdot\frac{\partial g}{\partial\mathbf{P}},

that is

\displaystyle  \left\{ f,g\right\} =\sum_{j=1}^{n}\frac{\partial f}{\partial p_{j}}\frac{\partial g}{\partial x_{j}}-\frac{\partial f}{\partial x_{j}}\frac{\partial g}{\partial p_{j}}

where {\mathbf{X}=\left(x_{1},\ldots,x_{n}\right)} are generalized coordinates and {\mathbf{P}=\left(p_{1},\ldots,p_{n}\right)} are generalized momenta.

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