![m = 1, hbar = 1, V[0] = -15, alpha = `+`(`*`(`/`(1, 8), `*`(Pi)))](images_QM_Bind/03_QM_Bind_1.gif){kind=small}
Lösung der freien 1D-Schrödinger-Gleichung
für negative Energien
Parameter:
|
(1) |
1) Schrödinger-Gleichung mit Kasten-Potential
Schrödinger-Gleichung
 |
(1.1) |
Kasten-Potential:
![V = piecewise(`<`(abs(x), 1), V[0], 0)](images_QM_Bind/03_QM_Bind_3.gif){kind=small} |
(1.2) |
Potential-Plot
 |
|
 |
Schrödinger-Gleichung mit Kasten-Potential
![`*`(I, `*`(hbar, `*`(diff(psi(x, t), t)))) = `+`(`-`(`/`(`*`(`/`(1, 2), `*`(diff(diff(psi(x, t), x), x))), `*`(m))), `*`(piecewise(`<`(abs(x), 1), V[0], 0), `*`(psi(x, t))))](images_QM_Bind/03_QM_Bind_6.gif) |
(1.3) |
2) Zeitunabhängige Schrödinger-Gleichung (ZSG)
Ansatz zur Seperation der Variablen:
 |
(2.1) |
Einsetzen und durch psi teilen
 |
(2.2) |
also
 |
(2.3) |
die linke Seite hängt nicht vom Ort x ab, die rechte nicht von der Zeit t, also sind beide Seiten Konstant E (später erweist sich E als die Energie)
 |
(2.4) |
Zeitunabhängige Schrödinger-Gleichung:
![`+`(`-`(`/`(`*`(`/`(1, 2), `*`(diff(diff(u(x), x), x))), `*`(m))), `*`(piecewise(`<`(abs(x), 1), V[0], 0), `*`(u(x)))) = `*`(u(x), `*`(E))](images_QM_Bind/03_QM_Bind_11.gif) |
(2.5) |
3) Symmetrische Quantenfelder
3.1) Ansatz
Mit Normierungskonstanten Fs_Nor:
 = `*`(Fs[Nor], `*`(piecewise(`<`(abs(x), 1), cos(`*`(k, `*`(x))), `*`(A[s], `*`(exp(`+`(`-`(`*`(kappa, `*`(abs(x)))))))))))](images_QM_Bind/03_QM_Bind_12.gif){kind=small} |
(3.1.1) |
Einsetzen ind ZSG
![`+`(`-`(`/`(`*`(`/`(1, 2), `*`(diff(diff(`*`(Fs[Nor], `*`(piecewise(`<`(abs(x), 1), cos(`*`(k, `*`(x))), `*`(A[s], `*`(exp(`+`(`-`(`*`(kappa, `*`(abs(x))))))))))), x), x))), `*`(m))), `*`(piecewise(`<...](images_QM_Bind/03_QM_Bind_13.gif) ![`+`(`-`(`/`(`*`(`/`(1, 2), `*`(diff(diff(`*`(Fs[Nor], `*`(piecewise(`<`(abs(x), 1), cos(`*`(k, `*`(x))), `*`(A[s], `*`(exp(`+`(`-`(`*`(kappa, `*`(abs(x))))))))))), x), x))), `*`(m))), `*`(piecewise(`<...](images_QM_Bind/03_QM_Bind_14.gif){kind=small} |
(3.1.2) |
für x>1:
![`+`(`-`(`/`(`*`(`/`(1, 2), `*`(Fs[Nor], `*`(A[s], `*`(`^`(kappa, 2), `*`(exp(`+`(`-`(`*`(kappa, `*`(x)))))))))), `*`(m)))) = `*`(Fs[Nor], `*`(A[s], `*`(exp(`+`(`-`(`*`(kappa, `*`(x))))), `*`(E))))](images_QM_Bind/03_QM_Bind_15.gif){kind=small} |
(3.1.3) |
also
 |
(3.1.4) |
für |x|<1:
![`+`(`/`(`*`(`/`(1, 2), `*`(Fs[Nor], `*`(cos(`*`(k, `*`(x))), `*`(`^`(k, 2))))), `*`(m)), `*`(V[0], `*`(Fs[Nor], `*`(cos(`*`(k, `*`(x))))))) = `*`(Fs[Nor], `*`(cos(`*`(k, `*`(x))), `*`(E)))](images_QM_Bind/03_QM_Bind_17.gif){kind=small} |
(3.1.5) |
also
![k = `*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)))](images_QM_Bind/03_QM_Bind_18.gif){kind=small} |
(3.1.6) |
insgesammt
 = `*`(Fs[Nor], `*`(piecewise(`<`(abs(x), 1), cos(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)), `*`(x))), `*`(A[s], `*`(exp(`+`(`-`(`*`(`^`(`+`(`-`(`*`(2, `*`...](images_QM_Bind/03_QM_Bind_19.gif) |
(3.1.7) |
3.2) Stetigkeit bei x=1
Bedingung:
, x = 1, left) = limit(u[s](x), x = 1, right)](images_QM_Bind/03_QM_Bind_20.gif){kind=small} |
(3.2.1) |
Mit Quantenfeld
![limit(`*`(Fs[Nor], `*`(piecewise(`<`(abs(x), 1), cos(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)), `*`(x))), `*`(A[s], `*`(exp(`+`(`-`(`*`(`^`(`+`(`-`(`*`(2, `*`(E, ...](images_QM_Bind/03_QM_Bind_21.gif) ![limit(`*`(Fs[Nor], `*`(piecewise(`<`(abs(x), 1), cos(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)), `*`(x))), `*`(A[s], `*`(exp(`+`(`-`(`*`(`^`(`+`(`-`(`*`(2, `*`(E, ...](images_QM_Bind/03_QM_Bind_22.gif) |
(3.2.2) |
Limes ausführen:
![`*`(cos(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)))), `*`(Fs[Nor])) = `/`(`*`(Fs[Nor], `*`(A[s])), `*`(exp(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2))))))](images_QM_Bind/03_QM_Bind_23.gif){kind=small} |
(3.2.3) |
Auflösen
![A[s] = `*`(cos(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)))), `*`(exp(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2))))))](images_QM_Bind/03_QM_Bind_24.gif){kind=small} |
(3.2.4) |
insgesamt
 = `*`(Fs[Nor], `*`(piecewise(`<`(abs(x), 1), cos(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)), `*`(x))), `*`(cos(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), ...](images_QM_Bind/03_QM_Bind_25.gif) |
(3.2.5) |
3.3) Differenzierbarkeit bei x=1
Bedingung
, x), x = 1, left) = limit(diff(u[s](x), x), x = 1, right)](images_QM_Bind/03_QM_Bind_26.gif){kind=small} |
(3.3.1) |
Mit Quantenfeld
![`+`(`-`(`*`(sin(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)))), `*`(Fs[Nor], `*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2))))))) = `...](images_QM_Bind/03_QM_Bind_27.gif){kind=small} ![`+`(`-`(`*`(sin(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)))), `*`(Fs[Nor], `*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2))))))) = `...](images_QM_Bind/03_QM_Bind_28.gif){kind=small} |
(3.3.2) |
Gleichung für Differenzierbarkeit
 |
(3.3.3) |
linke Seite als Funktion von E, die Nullstellen sind die Lösung letzten Gleichung
 |
Nullstellen numerisch
![E[s1] = -14.12055674, E[s2] = -7.333026712](images_QM_Bind/03_QM_Bind_31.gif){kind=small} |
(3.3.4) |
3.4 Normierung
Normierung
, 2)), x = `+`(`-`(infinity)) .. infinity)](images_QM_Bind/03_QM_Bind_32.gif){kind=small} |
(3.4.1) |
einsetzen
![1 = int(`*`(`^`(Fs[Nor], 2), `*`(`^`(piecewise(`<`(abs(x), 1), cos(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)), `*`(x))), `*`(cos(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `...](images_QM_Bind/03_QM_Bind_33.gif) |
(3.4.2) |
Integrieren und Auflösen
![Fs[Nor] = `/`(`*`(`^`(`*`(`+`(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2)), `*`(`^`(exp(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2))))), 2), `*`(cos(`*`(`^`(`+`(`-`(`*`(...](images_QM_Bind/03_QM_Bind_34.gif){kind=small} ![Fs[Nor] = `/`(`*`(`^`(`*`(`+`(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2)), `*`(`^`(exp(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2))))), 2), `*`(cos(`*`(`^`(`+`(`-`(`*`(...](images_QM_Bind/03_QM_Bind_35.gif){kind=small} ![Fs[Nor] = `/`(`*`(`^`(`*`(`+`(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2)), `*`(`^`(exp(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2))))), 2), `*`(cos(`*`(`^`(`+`(`-`(`*`(...](images_QM_Bind/03_QM_Bind_36.gif){kind=small} ![Fs[Nor] = `/`(`*`(`^`(`*`(`+`(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2)), `*`(`^`(exp(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2))))), 2), `*`(cos(`*`(`^`(`+`(`-`(`*`(...](images_QM_Bind/03_QM_Bind_37.gif){kind=small} ![Fs[Nor] = `/`(`*`(`^`(`*`(`+`(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2)), `*`(`^`(exp(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2))))), 2), `*`(cos(`*`(`^`(`+`(`-`(`*`(...](images_QM_Bind/03_QM_Bind_38.gif){kind=small} ![Fs[Nor] = `/`(`*`(`^`(`*`(`+`(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2)), `*`(`^`(exp(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2))))), 2), `*`(cos(`*`(`^`(`+`(`-`(`*`(...](images_QM_Bind/03_QM_Bind_39.gif){kind=small} ![Fs[Nor] = `/`(`*`(`^`(`*`(`+`(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2)), `*`(`^`(exp(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2))))), 2), `*`(cos(`*`(`^`(`+`(`-`(`*`(...](images_QM_Bind/03_QM_Bind_40.gif){kind=small} ![Fs[Nor] = `/`(`*`(`^`(`*`(`+`(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2)), `*`(`^`(exp(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2))))), 2), `*`(cos(`*`(`^`(`+`(`-`(`*`(...](images_QM_Bind/03_QM_Bind_41.gif){kind=small} |
(3.4.3) |
Also sind die beiden ersten symmetrischen Lösungen
 = piecewise(`<=`(x, -1.), `+`(`*`(45.14, `*`(exp(`+`(`*`(5.314, `*`(x))))))), `<`(x, 1.), `+`(`*`(.9174, `*`(cos(`+`(`*`(1.326, `*`(x))))))), `<=`(1., x), `+`(`*`(45.14, `*`(exp(`+`(`-`(`*`(5...](images_QM_Bind/03_QM_Bind_42.gif) |
(3.4.4) |
4) Antisymmetrische Quantenfelder
4.1) Ansatz
Mit Normierungskonstanten Fa_Nor
 = `*`(Fa[Nor], `*`(piecewise(`<`(abs(x), 1), sin(`*`(k, `*`(x))), `/`(`*`(x, `*`(A[a], `*`(exp(`+`(`-`(`*`(kappa, `*`(abs(x))))))))), `*`(abs(x))))))](images_QM_Bind/03_QM_Bind_43.gif) |
(4.1.1) |
Einsetzen ind ZSG
![`+`(`-`(`/`(`*`(`/`(1, 2), `*`(diff(diff(`*`(Fa[Nor], `*`(piecewise(`<`(abs(x), 1), sin(`*`(k, `*`(x))), `/`(`*`(x, `*`(A[a], `*`(exp(`+`(`-`(`*`(kappa, `*`(abs(x))))))))), `*`(abs(x)))))), x), x))), ...](images_QM_Bind/03_QM_Bind_44.gif) ![`+`(`-`(`/`(`*`(`/`(1, 2), `*`(diff(diff(`*`(Fa[Nor], `*`(piecewise(`<`(abs(x), 1), sin(`*`(k, `*`(x))), `/`(`*`(x, `*`(A[a], `*`(exp(`+`(`-`(`*`(kappa, `*`(abs(x))))))))), `*`(abs(x)))))), x), x))), ...](images_QM_Bind/03_QM_Bind_45.gif) |
(4.1.2) |
Ansatz soll ZSG lösen! - x>1:
![`+`(`-`(`/`(`*`(`/`(1, 2), `*`(Fa[Nor], `*`(A[a], `*`(`^`(kappa, 2), `*`(exp(`+`(`-`(`*`(kappa, `*`(x)))))))))), `*`(m)))) = `*`(Fa[Nor], `*`(A[a], `*`(exp(`+`(`-`(`*`(kappa, `*`(x))))), `*`(E))))](images_QM_Bind/03_QM_Bind_46.gif){kind=small} |
(4.1.3) |
auflösen
 |
(4.1.4) |
|x|<1
![`+`(`/`(`*`(`/`(1, 2), `*`(Fa[Nor], `*`(sin(`*`(k, `*`(x))), `*`(`^`(k, 2))))), `*`(m)), `*`(V[0], `*`(Fa[Nor], `*`(sin(`*`(k, `*`(x))))))) = `*`(Fa[Nor], `*`(sin(`*`(k, `*`(x))), `*`(E)))](images_QM_Bind/03_QM_Bind_48.gif){kind=small} |
(4.1.5) |
auflösen
![k = `*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)))](images_QM_Bind/03_QM_Bind_49.gif){kind=small} |
(4.1.6) |
also:
 = `*`(Fa[Nor], `*`(piecewise(`<`(abs(x), 1), sin(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)), `*`(x))), `/`(`*`(x, `*`(A[a], `*`(exp(`+`(`-`(`*`(`^`(`+`(`-`...](images_QM_Bind/03_QM_Bind_50.gif) |
(4.1.7) |
4.2) Stetigkeit bei x=1
Bedingung.
, x = 1, left) = limit(u[a](x), x = 1, right)](images_QM_Bind/03_QM_Bind_51.gif){kind=small} |
(4.2.1) |
Mit Quantenfeld
![`*`(sin(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)))), `*`(Fa[Nor])) = `/`(`*`(Fa[Nor], `*`(A[a])), `*`(exp(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2))))))](images_QM_Bind/03_QM_Bind_52.gif){kind=small} |
(4.2.2) |
auflösen
![A[a] = `*`(sin(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)))), `*`(exp(`*`(`^`(`+`(`-`(`*`(2, `*`(E, `*`(m))))), `/`(1, 2))))))](images_QM_Bind/03_QM_Bind_53.gif){kind=small} |
(4.2.3) |
also:
 = `*`(Fa[Nor], `*`(piecewise(`<`(abs(x), 1), sin(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)), `*`(x))), `/`(`*`(x, `*`(sin(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], ...](images_QM_Bind/03_QM_Bind_54.gif) |
(4.2.4) |
4.3) Differenzierbarkeit bei x=1
Bedingung
, x), x = 1, left) = limit(diff(u[a](x), x), x = 1, right)](images_QM_Bind/03_QM_Bind_55.gif){kind=small} |
(4.3.1) |
Mit Quantenfeld
![`*`(cos(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)))), `*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)))) = `+`(`-`(`*`(sin(`*`(`^`(`...](images_QM_Bind/03_QM_Bind_56.gif){kind=small} |
(4.3.2) |
Gleichung für Differenzierbarkeit
 |
(4.3.3) |
Graphische Darstellung: Nullstellen sind Lösungen
 |
Nullstellen numerisch
![E[a1] = -11.51812382, E[a2] = -2.043684011](images_QM_Bind/03_QM_Bind_59.gif){kind=small} |
(4.3.4) |
4.4) Normierung
Normierung
, 2)), x = `+`(`-`(infinity)) .. infinity)](images_QM_Bind/03_QM_Bind_60.gif){kind=small} |
(4.4.1) |
einsetzen
![1 = int(`*`(`^`(Fa[Nor], 2), `*`(`^`(piecewise(`<`(abs(x), 1), sin(`*`(`^`(`+`(`-`(`*`(2, `*`(V[0], `*`(m)))), `*`(2, `*`(E, `*`(m)))), `/`(1, 2)), `*`(x))), `/`(`*`(x, `*`(sin(`*`(`^`(`+`(`-`(`*`(2, ...](images_QM_Bind/03_QM_Bind_61.gif) |
(4.4.2) |
integrieren und auflösen
![Fa[Nor] = `/`(`*`(`^`(`+`(`-`(`*`(`+`(`*`(2, `*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2)), `*`(cos(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`*`(m, `*`(`+`(`-`(V[0]), E))), `/`(1, 2))))), ...](images_QM_Bind/03_QM_Bind_62.gif){kind=small} ![Fa[Nor] = `/`(`*`(`^`(`+`(`-`(`*`(`+`(`*`(2, `*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2)), `*`(cos(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`*`(m, `*`(`+`(`-`(V[0]), E))), `/`(1, 2))))), ...](images_QM_Bind/03_QM_Bind_63.gif){kind=small} ![Fa[Nor] = `/`(`*`(`^`(`+`(`-`(`*`(`+`(`*`(2, `*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2)), `*`(cos(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`*`(m, `*`(`+`(`-`(V[0]), E))), `/`(1, 2))))), ...](images_QM_Bind/03_QM_Bind_64.gif){kind=small} ![Fa[Nor] = `/`(`*`(`^`(`+`(`-`(`*`(`+`(`*`(2, `*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2)), `*`(cos(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`*`(m, `*`(`+`(`-`(V[0]), E))), `/`(1, 2))))), ...](images_QM_Bind/03_QM_Bind_65.gif){kind=small} ![Fa[Nor] = `/`(`*`(`^`(`+`(`-`(`*`(`+`(`*`(2, `*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2)), `*`(cos(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`*`(m, `*`(`+`(`-`(V[0]), E))), `/`(1, 2))))), ...](images_QM_Bind/03_QM_Bind_66.gif){kind=small} ![Fa[Nor] = `/`(`*`(`^`(`+`(`-`(`*`(`+`(`*`(2, `*`(`^`(2, `/`(1, 2)), `*`(`^`(`+`(`-`(`*`(E, `*`(m)))), `/`(1, 2)), `*`(cos(`*`(`^`(2, `/`(1, 2)), `*`(`^`(`*`(m, `*`(`+`(`-`(V[0]), E))), `/`(1, 2))))), ...](images_QM_Bind/03_QM_Bind_67.gif){kind=small} |
(4.4.3) |
Die niedersten zwei Lösungen
 = piecewise(`<=`(x, -1.), `+`(`*`(53.24, `*`(exp(`+`(`*`(4.800, `*`(x))))))), `<`(x, 1.), `+`(`-`(`*`(.9097, `*`(sin(`+`(`*`(2.639, `*`(x)))))))), `<=`(1., x), `+`(`-`(`*`(53.24, `*`(exp(`+`(...](images_QM_Bind/03_QM_Bind_68.gif) |
(4.4.4) |
5) Plot Quantenfelder
Die niedersten 4 Lösungen
symmetrisch
 = piecewise(`<=`(x, -1.), `+`(`*`(45.14, `*`(exp(`+`(`*`(5.314, `*`(x))))))), `<`(x, 1.), `+`(`*`(.9174, `*`(cos(`+`(`*`(1.326, `*`(x))))))), `<=`(1., x), `+`(`*`(45.14, `*`(exp(`+`(`-`(`*`(5...](images_QM_Bind/03_QM_Bind_69.gif) |
(5.1) |
antisymmetrisch
 = piecewise(`<=`(x, -1.), `+`(`*`(53.24, `*`(exp(`+`(`*`(4.800, `*`(x))))))), `<`(x, 1.), `+`(`-`(`*`(.9097, `*`(sin(`+`(`*`(2.639, `*`(x)))))))), `<=`(1., x), `+`(`-`(`*`(53.24, `*`(exp(`+`(...](images_QM_Bind/03_QM_Bind_70.gif) |
(5.2) |
Graphische Darstellung der niedersten vier Lösungen
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6) Energiespektrum
Parameter
![m = 1, hbar = 1, V[0] = -15, alpha = `+`(`*`(`/`(1, 8), `*`(Pi)))](images_QM_Bind/03_QM_Bind_75.gif){kind=small} |
(6.1) |
Energien
![E[s1] = -14.12055674, E[s2] = -7.333026712, E[a1] = -11.51812382, E[a2] = -2.043684011](images_QM_Bind/03_QM_Bind_76.gif){kind=small} |
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7) Zeitabhängigkeit der Wahrscheinlichkeitsdichte
Quantenfeld
, `*`(exp(`+`(`-`(`/`(`*`(`+`(I), `*`(E[s1], `*`(t))), `*`(hbar)))))))), `*`(sin(alpha), `*`(u[a1](x), `*`(exp(`+`(`-`(`/`(`*`(`+`(I), `*`(E[a1], `*`(t))), `*`(hb...](images_QM_Bind/03_QM_Bind_78.gif){kind=small} |
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(7.1) |
Dichte
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(7.2) |
Periode
![T_P = `+`(`/`(`*`(2., `*`(Pi)), `*`(`+`(E[a1], `-`(E[s1])))))](images_QM_Bind/03_QM_Bind_82.gif){kind=small} |
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(7.3) |
Dichte zu verschiedenen Zeiten
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8) Animation der Überlagerung aus Abschnitt 7)
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