Coordinate representation (quantum mechanics)

Coordinate representation (quantum mechanics) is a representation of quantum mechanics operators in which the operators and the wave function depend on spatial coordinates. In this representation, the coordinate operator is diagonal.

Content

Schrödinger equation

In this representation, the Schrödinger equation has the form:

$(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(r))\psi =i{\hbar }{\frac {\partial \psi }{\partial t}}$ ${\ displaystyle (- {\ frac {\ hbar ^ {2}} {2m}} \ nabla ^ {2} + V (r)) \ psi = i {\ hbar} {\ frac {\ partial \ psi} { \ partial t}}}$

- time dependent, and

$(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(r))\psi =E\psi$ ${\ displaystyle (- {\ frac {\ hbar ^ {2}} {2m}} \ nabla ^ {2} + V (r)) \ psi = E \ psi}$

time-independent (everywhere r is the radius vector of the point where the wave function is taken).

Some operators in the coordinate representation

$r$ ${\ displaystyle r}$ -coordinate;

$-i{\hbar }{\frac {\partial }{\partial r}}$ ${\ displaystyle -i {\ hbar} {\ frac {\ partial} {\ partial r}}}$ - impulse ;

$-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(r)$ ${\ displaystyle - {\ frac {\ hbar ^ {2}} {2m}} \ nabla ^ {2} + V (r)}$ - Hamiltonian .

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To go into impulse presentation, you need either

1) Solve the problem in the coordinate and go to the impulse using the superposition relation

$\psi (p)={\frac {1}{\sqrt {2\pi {\hbar }}}}\int \psi (x)e^{-ipx/\hbar }dx$ ${\ displaystyle \ psi (p) = {\ frac {1} {\ sqrt {2 \ pi {\ hbar}}} \ int \ psi (x) e ^ {- ipx / \ hbar} dx}$

PS The transition back to the coordinate representation can be written as $\psi (x)={\frac {1}{\sqrt {2\pi {\hbar }}}}\int \psi (p)e^{ipx/\hbar }dp$ ${\ displaystyle \ psi (x) = {\ frac {1} {\ sqrt {2 \ pi {\ hbar}}}} int \ psi (p) e ^ {ipx / \ hbar} dp}$

It is easy to see that this is the direct and inverse Fourier transforms . In three-dimensional space, the integral multiplier must be replaced by ${\frac {1}{\sqrt {(2\pi \hbar )^{3}}}}$ ${\ displaystyle {\ frac {1} {\ sqrt {(2 \ pi \ hbar) ^ {3}}}}}$

2) Change the Hamiltonian to ${\hat {H}}={\frac {p^{2}}{2m}}+V(i{\hbar }{\frac {\partial }{\partial p}})$ ${\ displaystyle {\ hat {H}} = {\ frac {p ^ {2}} {2m}} + V (i {\ hbar} {\ frac {\ partial} {\ partial p}})}$ and solve a problem with him.

Literature

Tarasov L.V. Fundamentals of quantum mechanics. M.: Book House "LIBROCOM", 2014.