Endless pit

A pit with infinite walls , in quantum mechanics , is a model of a particle enclosed in a “box” of a certain shape. In the one-dimensional case, this box is a finite segment . Inside the segment, the potential is considered to be zero. At all other points of the real line, the potential goes to infinity. Mathematically, this is usually reflected in the boundary conditions, assuming that the wave functions vanish at the ends of the segment. This potential is the limiting case of a rectangular quantum well . In the multidimensional case, the potential is considered equal to zero inside a certain region at the boundaries of which the Dirichlet boundary conditions are set. Often consider a rectangular area (rectangular "box").

One-dimensional potential well with infinite walls

The potential of a one-dimensional potential well with infinite walls has the form

U(x)={\begin{cases}0,&x\in (-{\frac {a}{2}},{\frac {a}{2}}),\\\infty ,&x\notin (-{\frac {a}{2}},{\frac {a}{2}})\end{cases}}

{\ displaystyle U (x) = {\ begin {cases} 0, & x \ in (- {\ frac {a} {2}}, {\ frac {a} {2}}), \\\ infty, & x \ notin (- {\ frac {a} {2}}, {\ frac {a} {2}}) \ end {cases}}}

The stationary Schrödinger equation on the interval $\left(-{\frac {a}{2}},{\frac {a}{2}}\right)$ ${\ displaystyle \ left (- {\ frac {a} {2}}, {\ frac {a} {2}} \ right)}$

-{\frac {\hbar ^{2}}{2m}}\Psi ''(x)=E\Psi (x).

{\ displaystyle - {\ frac {\ hbar ^ {2}} {2m}} \ Psi '' (x) = E \ Psi (x).}

With designation $k={\sqrt {2mE/\hbar ^{2}}}$ ${\ displaystyle k = {\ sqrt {2mE / \ hbar ^ {2}}}}$ , it will take the form:

\Psi ''(x)+k^{2}\Psi (x)=0.

{\ displaystyle \ Psi '' (x) + k ^ {2} \ Psi (x) = 0.}

It is convenient to present the general solution in the form of a linear shell of even and odd functions:

\Psi (x)=C^{+}\cos kx+C^{-}\sin kx.

{\ displaystyle \ Psi (x) = C ^ {+} \ cos kx + C ^ {-} \ sin kx.}

The boundary values are:

\Psi \left(-{\frac {a}{2}}\right)=\Psi \left({\frac {a}{2}}\right)=0.

{\ displaystyle \ Psi \ left (- {\ frac {a} {2}} \ right) = \ Psi \ left ({\ frac {a} {2}} \ right) = 0.}

They lead to a homogeneous system of linear equations:

{\begin{cases}C^{+}\cos {\frac {ka}{2}}+C^{-}\sin {\frac {ka}{2}}=0,\\C^{+}\cos {\frac {ka}{2}}-C^{-}\sin {\frac {ka}{2}}=0,\end{cases}}

{\ displaystyle {\ begin {cases} C ^ {+} \ cos {\ frac {ka} {2}} + C ^ {-} \ sin {\ frac {ka} {2}} = 0, \\ C ^ {+} \ cos {\ frac {ka} {2}} - C ^ {-} \ sin {\ frac {ka} {2}} = 0, \ end {cases}}}

which has non-trivial solutions provided that its determinant is equal to zero:

-2\cos {\frac {ka}{2}}\sin {\frac {ka}{2}}=0,

{\ displaystyle -2 \ cos {\ frac {ka} {2}} \ sin {\ frac {ka} {2}} = 0,}

that after trigonometric transformations takes the form:

\sin ka=0.

{\ displaystyle \ sin ka = 0.}

The roots of this equation are of the form

k_{n}={\frac {\pi n}{a}},\qquad n\in \mathbb {Z} _{+}.

{\ displaystyle k_ {n} = {\ frac {\ pi n} {a}}, \ qquad n \ in \ mathbb {Z} _ {+}.}

Substituting into the system, we have:

C_{n}^{-}=0,\qquad n=2n_{0}+1,\qquad n_{0}\in \mathbb {Z} _{+},

{\ displaystyle C_ {n} ^ {-} = 0, \ qquad n = 2n_ {0} +1, \ qquad n_ {0} \ in \ mathbb {Z} _ {+},}

C_{n}^{+}=0,\qquad n=2n_{0},\qquad n_{0}\in \mathbb {Z} _{+}.

{\ displaystyle C_ {n} ^ {+} = 0, \ qquad n = 2n_ {0}, \ qquad n_ {0} \ in \ mathbb {Z} _ {+}.}

Thus, the solutions fall into two series - even and odd solutions:

\Psi _{n_{0}}^{+}(x)=C_{2n_{0}+1}^{+}\cos {\frac {(2n_{0}+1)\pi x}{a}},\qquad n_{0}\in \mathbb {Z} _{+},

{\ displaystyle \ Psi _ {n_ {0}} ^ {+} (x) = C_ {2n_ {0} +1} ^ {+} \ cos {\ frac {(2n_ {0} +1) \ pi x } {a}}, \ qquad n_ {0} \ in \ mathbb {Z} _ {+},}

\Psi _{n_{0}}^{-}(x)=C_{2n_{0}}^{-}\sin {\frac {2n_{0}\pi x}{a}},\qquad n_{0}\in \mathbb {Z} _{+}.

{\ displaystyle \ Psi _ {n_ {0}} ^ {-} (x) = C_ {2n_ {0}} ^ {-} \ sin {\ frac {2n_ {0} \ pi x} {a}}, \ qquad n_ {0} \ in \ mathbb {Z} _ {+}.}

The fact that decisions are divided into even and odd is due to the fact that the potential itself is an even function. Taking into account normalization

\int \limits _{-{\frac {a}{2}}}^{\frac {a}{2}}\left(\Psi _{n_{0}}^{\pm }(x)\right)^{2}dx=1,

{\ displaystyle \ int \ limits _ {- {\ frac {a} {2}}} ^ {\ frac {a} {2}} \ left (\ Psi _ {n_ {0}} ^ {\ pm} ( x) \ right) ^ {2} dx = 1,}

we get the explicit form of the normalization factors:

C_{2n_{0}+1}^{+}=C_{2n_{0}}^{-}={\sqrt {\frac {2}{a}}}.

{\ displaystyle C_ {2n_ {0} +1} ^ {+} = C_ {2n_ {0}} ^ {-} = {\ sqrt {\ frac {2} {a}}}.}

As a result, we obtain the eigenfunctions of the Hamiltonian :

\Psi _{n_{0}}^{+}(x)={\sqrt {\frac {2}{a}}}\cos {\frac {(2n_{0}+1)\pi x}{a}},\qquad n_{0}\in \mathbb {Z} _{+},

{\ displaystyle \ Psi _ {n_ {0}} ^ {+} (x) = {\ sqrt {\ frac {2} {a}}} \ cos {\ frac {(2n_ {0} +1) \ pi x} {a}}, \ qquad n_ {0} \ in \ mathbb {Z} _ {+},}

\Psi _{n_{0}}^{-}(x)={\sqrt {\frac {2}{a}}}\sin {\frac {2n_{0}\pi x}{a}},\qquad n_{0}\in \mathbb {Z} _{+},

{\ displaystyle \ Psi _ {n_ {0}} ^ {-} (x) = {\ sqrt {\ frac {2} {a}}} \ sin {\ frac {2n_ {0} \ pi x} {a }}, \ qquad n_ {0} \ in \ mathbb {Z} _ {+},}

with the corresponding energy spectrum:

E_{n_{0}}^{+}={\frac {\hbar ^{2}\pi ^{2}(2n_{0}+1)^{2}}{2ma^{2}}}

{\ displaystyle E_ {n_ {0}} ^ {+} = {\ frac {\ hbar ^ {2} \ pi ^ {2} (2n_ {0} +1) ^ {2}} {2ma ^ {2} }}}

E_{n_{0}}^{-}={\frac {\hbar ^{2}\pi ^{2}(2n_{0})^{2}}{2ma^{2}}}

{\ displaystyle E_ {n_ {0}} ^ {-} = {\ frac {\ hbar ^ {2} \ pi ^ {2} (2n_ {0}) ^ {2}} {2ma ^ {2}}} }

Literature

Boom D. Quantum Theory. - Science, Main Edition of Physics and Mathematics, 1965.
Flygge Z. Problems in quantum mechanics. - Publishing house LCI, 2008. - T. 1.

Endless pit

One-dimensional potential well with infinite walls

Literature

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