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The wavefunction is a complex scalar field that describes a quantum mechanical system. The square of the modulus of the wave function gives the probability of the system to be found in a particular state.

In the Schrodinger Wave formulation of Quantum Mechanics, the wavefunction can be determined by the , which, in it's most general form, can be stated as:

$$\hat H\Psi=i\hbar\frac{\partial\Psi}{\partial t}$$

In the case of a "Euclidean Hamiltonian" given by the eigenvalue $E=\frac{p^2}{2m}+U$, this becomes:

$$\left(\frac{p^2}{2m}+U\right) \Psi=i\hbar\frac{\partial\Psi}{\partial t}$$

Since the momentum $p$ is interpreted as an eigenvalue of $-i\hbar\nabla$, i.e. $p\Psi=-i\hbar\nabla \Psi$,

$$\left(-\frac{\hbar^2}{2m}\nabla^2+U\right)\Psi= i\hbar\frac{\partial\Psi}{\partial t} $$

This is known as the time-independent Schrodinger's equation. Note, that as the Hamiltonian used was Euclidean, this equation is in fact, non - relativistic. The relativistic version of this equation in Relativistic Quantum Mechanics (and also in , but there it describes spin-1/2 fields) is the .

The wavefunction also appears in Feynman's formulation of Quantum Mechanics. In the Path Integral formulation, a functional, called the phase is associated with each path:

$$\phi = A e^\frac{iS}{\hbar} $$

The Kernel or the Matrix Element, is the path integral of this phase.

$$K(x ) =\int\phi\mbox{ } \mathcal{D}x $$

Where $x$ is actually the position of the particle in spacetime. The wavefunction, finally is given by:

$$\Psi(x)=\int_{-\infty}^\infty \left(K(x,x_0)\Psi(x_0) \right) \mbox{d} x_0 $$

It is often surprising to many that the absolute value of the phase squared, $|\phi|^2$, is constant for all paths, at $A^2$. However, this actually makes sense, as the position of the particle is initially completely well-defined, so Heisenberg's Uncertainty Principle tells us that we would have no idea about the momentum, and thus no idea about it's future position. However, the next moment, you know absolutely nothing about it's momentum, and so on. This process coarse-grains a particular path, the classical path, which means it is much more probable than the other paths.

The mathematical description of this can be obtained by standard procedures (c.f. Feynman, Hibbs, Styer "Quantum Mechanics and Path Integrals", pg 77 - 79) and the final result is the Schrodinger's Equation.

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