User:Espen180/Quantum Mechanics/Formulation of Quantum Mechanics

Review of Classical Mechanics

Classical mechanics is divided into two branches, Lagrangian mechanics and Hamiltonian mechanics. In Lagrangian mechanics, a system having $n$ degrees of freedom $q_{1},...,q_{n}$ is described by a function $L(q_{1},...,q_{n},{\dot {q_{1}}},...,{\dot {q_{n}}},t)\equiv L(q,{\dot {q}},t)$ of the degrees of fredom and their temporal derivatives. The function $L$ is called the Lagrangian of the system. The equations of motion of the system are given by Hamilton's principle, stating that the degrees of freedom change in such a way that the integral

$S[t_{1},t_{2}]=\int _{t_{1}}^{t_{2}}L(q,{\dot {q}},t)\mathrm {d} t$

is at an extremum with respect to the path. This is a problem in variational calculus which we will not discuss here. It's solution is

${\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q_{i}}}}}\right)-{\frac {\partial L}{\partial q_{i}}}=0$

for each individual index $i$. These equations are called the Euler-Lagrange equations. Thus we obtain $n$ second-order partial differential equations describing the system. This gives us $2n$ initial conditions which determine the evolution of the system. However, the Lagrangian formalism is not suited for quantum mechanics. We need the other formalism, Hamiltonian mechanics. The Hamiltonian formalism is based on the following fact. Assume that the degree of freedom $q_{i}$ does not appear in $L$ for some $i$. Then ${\frac {\partial L}{\partial q_{i}}}=0$, so we get

${\frac {\partial L}{\partial {\dot {q_{i}}}}}=k_{i}$

where $k_{i}$ is a constant. In other words, we get a conserved quantity. The hamiltonian formalism is based on replacing ${\dot {q_{i}}}$ in $L$ by $p_{i}={\frac {\partial L}{\partial {\dot {q_{i}}}}}$ for all $i$. We can do this by performing a Legendre transformation on $L$. $p_{i}$ is called the canonical or conjugate momentum associated with $q_{i}$. We define

$H=p_{i}{\dot {q_{i}}}-L$.

Solving $p_{i}={\frac {\partial L}{\partial {\dot {q_{i}}}}}$ for ${\dot {q_{i}}}$ and inserting, we get the Hamiltonian function $H(q,p,t)$. The equations of motion can be found from the Euler-Lagrange equations. We get

${\dot {q_{i}}}={\frac {\partial H}{\partial p_{i}}}$,

${\dot {p_{i}}}=-{\frac {\partial H}{\partial q_{i}}}$,

${\frac {\mathrm {d} H}{\mathrm {d} t}}=-{\frac {\mathrm {d} L}{\mathrm {d} t}}$.

These are called Hamilton's equations. We get $2n$ first-order equation describing our system, again giving us $2n$ initial conditions, as expected.

Poisson Brackets

Define the Poisson bracket as the expression

$\{A,B\}=\sum _{i=1}^{n}{\frac {\partial A}{\partial q_{i}}}{\frac {\partial B}{\partial p_{i}}}-{\frac {\partial A}{\partial p_{i}}}{\frac {\partial B}{\partial q_{i}}}$.

It is readily checked that

$\{q_{i},p_{j}\}=\delta _{ij}$

where $\delta _{ij}=1$ if $i=j$ and $0$ otherwise.

We can also obtain a useful expression about the time evolution of arbitrary quatities. Let $F(q,p,t)$ be any (differentiable) quantity. We then have

${\frac {\mathrm {d} F}{\mathrm {d} t}}={\frac {\partial F}{\partial t}}+\sum _{i=1}^{n}{\frac {\partial F}{\partial q_{i}}}{\dot {q_{i}}}+{\frac {\partial F}{\partial p_{i}}}{\dot {p_{i}}}$

by the chain rule. Insering for the time derivatives, we get

${\frac {\mathrm {d} F}{\mathrm {d} t}}={\frac {\partial F}{\partial t}}+\sum _{i=1}^{n}{\frac {\partial F}{\partial q_{i}}}{\frac {\partial H}{\partial p_{i}}}-{\frac {\partial F}{\partial p_{i}}}{\frac {\partial H}{\partial q_{i}}}={\frac {\partial F}{\partial t}}+\{F,H\}$.

The poisson brackets have an important counterpart in quantum mechanics, and are used as a starting point for the theory.

Formulation of Quantum Mechanics

In quantum mechanics is based on the following postulates:

1. To each state of a physical system there corresponds a state vector $|\Psi (t)\rangle$ in a complex Hilbert space. The state vector has length 1, meaning $\langle \Psi (t)|\Psi (t)\rangle =1$, and its time evoltution satisfies the Schrödinger equation

$i\hbar \partial _{t}|\Psi (t)\rangle ={\hat {H}}|\Psi (t)\rangle$

where ${\hat {H}}$ is the Hamiltonian operator of the system. We will get back to determining ${\hat {H}}$ for a given system.

2. To each physical observable $F$ there corresponds a linear operator ${\hat {F}}$ on the Hilbert space. The operators ${\hat {q_{i}}}$ and ${\hat {p_{i}}}$ for the generalized coordinates and momenta satisfy

$[{\hat {q_{i}}},{\hat {p_{j}}}]=i\hbar \delta _{ij}$.

3. The expectation value of an observable $F$ is $\langle F\rangle =\langle \Psi (t)|{\hat {F}}|\Psi (t)\rangle$.

4. The only possible results when measuring the observable $F$ are the eigenvalues $f_{n}$ of ${\hat {F}}$.