Und das ist die Sandbox
Mathematik-Hilfe [[1]]
3 2 k B ⋅ T = ⟨ E ⟩ {\displaystyle {\frac {3}{2}}\,k_{\text{B}}\cdot T=\langle E\rangle }
∑ k = 1 N k 2 {\displaystyle \sum _{k=1}^{N}k^{2}}
ℏ = h 2 π {\displaystyle \hbar ={\frac {h}{2\pi }}}
Δ x ⋅ Δ p ≥ ℏ 2 {\displaystyle \Delta x\cdot \Delta p\geq {\frac {\hbar }{2}}}
λ = h p x = h 2 m E kin {\displaystyle \lambda ={\frac {h}{p_{x}}}={\frac {h}{\sqrt {2mE_{\text{kin}}}}}}
Ψ p x ( x , t ) = A ⋅ sin ( 2 π x λ − 2 π t T ) + B ⋅ cos ( 2 π x λ − 2 π t T ) {\displaystyle \Psi _{p_{x}}(x,t)=A\cdot {\text{sin}}\left({\frac {2\pi x}{\lambda }}-{\frac {2\pi t}{T}}\right)+B\cdot {\text{cos}}\left({\frac {2\pi x}{\lambda }}-{\frac {2\pi t}{T}}\right)}
Ψ E kin ( x , t ) = A ⋅ sin ( 2 m E kin ℏ ⋅ x − 2 π t T ) + B ⋅ cos ( 2 m E kin ℏ ⋅ x − 2 π t T ) {\displaystyle \Psi _{E_{\text{kin}}}(x,t)=A\cdot {\text{sin}}\left({\frac {\sqrt {2mE_{\text{kin}}}}{\hbar }}\cdot x-{\frac {2\pi t}{T}}\right)+B\cdot {\text{cos}}\left({\frac {\sqrt {2mE_{\text{kin}}}}{\hbar }}\cdot x-{\frac {2\pi t}{T}}\right)}
Ψ p A ( x , t ) = A ⋅ sin ( p x ℏ ⋅ x − 2 π t T ) + B ⋅ cos ( p x ℏ ⋅ x − 2 π t T ) {\displaystyle \Psi _{p_{A}}(x,t)=A\cdot {\text{sin}}\left({\frac {p_{x}}{\hbar }}\cdot x-{\frac {2\pi t}{T}}\right)+B\cdot {\text{cos}}\left({\frac {p_{x}}{\hbar }}\cdot x-{\frac {2\pi t}{T}}\right)}
Ψ ′ ( x ) = d Ψ ( x ) d x {\displaystyle \Psi '(x)={\frac {{\text{d}}\Psi (x)}{{\text{d}}x}}}
− ℏ 2 2 m d 2 d x 2 Ψ E kin ( x ) = E kin ⋅ Ψ E kin ( x ) {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}\Psi _{E_{\text{kin}}}(x)=E_{\text{kin}}\cdot \Psi _{E_{\text{kin}}}(x)}
E ^ kin = − ℏ 2 2 m d 2 d x 2 {\displaystyle {\hat {E}}_{\text{kin}}=-{\frac {\hbar ^{2}}{2m}}{\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}}
E ^ kin ⋅ Ψ kin ( x ) = E kin ⋅ Ψ kin ( x ) {\displaystyle {\hat {E}}_{\text{kin}}\cdot \Psi _{\text{kin}}(x)=E_{\text{kin}}\cdot \Psi _{\text{kin}}(x)}
Ψ ′ ( x ) = d Ψ ( x ) d x {\displaystyle \Psi '(x)={\frac {d\Psi (x)}{dx}}}
Ψ Gau β ∼ e − ( x − x 0 ) 2 2 σ 2 {\displaystyle \Psi _{{\text{Gau}}\beta }\sim e^{-{\frac {(x-x_{0})^{2}}{2\sigma ^{2}}}}}
E ^ kin ⋅ Ψ Gau β ( x ) = K o n s t a n t e ⋅ Ψ Gau β ( x ) {\displaystyle {\hat {E}}_{\text{kin}}\cdot \Psi _{{\text{Gau}}\beta }(x)=Konstante\cdot \Psi _{{\text{Gau}}\beta }(x)}
p = h λ {\displaystyle p={\frac {h}{\lambda }}}
λ = h p {\displaystyle \lambda ={\frac {h}{p}}}
Ψ = Ψ 1 ( x ) + Ψ 2 ( x ) {\displaystyle \Psi =\Psi _{1}(x)+\Psi _{2}\left(x\right)}
P ( x ) = Ψ 1 ( x ) + Ψ 2 ( x ) {\displaystyle P(x)=\Psi _{1}(x)+\Psi _{2}(x)}
P ( x ) = | Ψ 1 ( x ) + Ψ 2 ( x ) | 2 {\displaystyle P(x)=\left\vert \Psi _{1}(x)+\Psi _{2}(x)\right\vert ^{2}}
P ( x ) = | Ψ 1 ( x ) | 2 + | Ψ 2 ( x ) | 2 + 2 ⋅ Ψ 1 ( x ) ⋅ Ψ 2 ( x ) {\displaystyle P(x)=\left\vert \Psi _{1}(x)\right\vert ^{2}+\left\vert \Psi _{2}(x)\right\vert ^{2}+2\cdot \Psi _{1}(x)\cdot \Psi _{2}(x)}
P ( x ) ≠ P 1 ( x ) + P 2 ( x ) {\displaystyle P(x)\neq P_{1}(x)+P_{2}(x)}
P ( x ) = | Ψ 1 ( x ) | 2 + | Ψ 2 ( x ) | 2 + Ψ 1 ( x ) ⋅ Ψ 2 ⋇ ( x ) + Ψ 1 ⋇ ( x ) ⋅ Ψ 2 ( x ) {\displaystyle P(x)=\left\vert \Psi _{1}(x)\right\vert ^{2}+\left\vert \Psi _{2}(x)\right\vert ^{2}+\Psi _{1}(x)\cdot \Psi _{2}^{\divideontimes }(x)+\Psi _{1}^{\divideontimes }(x)\cdot \Psi _{2}(x)}
P ( x ) ⋅ Δ x = | Ψ ( x ) | 2 ⋅ Δ x {\displaystyle P(x)\cdot \Delta x=\left\vert \Psi (x)\right\vert ^{2}\cdot \Delta x}
Δ y ⋅ Δ p y ≈ h {\displaystyle \Delta y\cdot \Delta p_{y}\approx h}
Δ y ⋅ Δ p y ≥ h 4 π {\displaystyle \Delta y\cdot \Delta p_{y}\geq {\frac {h}{4\pi }}}
E kin ( I I I ) {\displaystyle E_{\text{kin}}^{(III)}}
Ψ E ges ( x , t ) = A ⋅ sin ( 2 m ( E ges − V 0 ) ℏ ⋅ x − 2 π t T ) + B ⋅ cos ( 2 m ( E ges − V 0 ) ℏ ⋅ x − 2 π t T ) {\displaystyle \Psi _{E_{\text{ges}}}(x,t)=A\cdot {\text{sin}}\left({\frac {\sqrt {2m(E_{\text{ges}}-V_{0})}}{\hbar }}\cdot x-{\frac {2\pi t}{T}}\right)+B\cdot {\text{cos}}\left({\frac {\sqrt {2m(E_{\text{ges}}-V_{0})}}{\hbar }}\cdot x-{\frac {2\pi t}{T}}\right)}
E ^ ges = − ℏ 2 2 m d 2 d x 2 + V ( x ) {\displaystyle {\hat {E}}_{\text{ges}}=-{\frac {\hbar ^{2}}{2m}}{\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}+V(x)}
E ^ ges = E ^ kin + E ^ pot {\displaystyle {\hat {E}}_{\text{ges}}={\hat {E}}_{\text{kin}}+{\hat {E}}_{\text{pot}}}
[ − ℏ 2 2 m d 2 d x 2 + V ( x ) ] Ψ ( x ) = E ges ⋅ Ψ ( x ) {\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}{\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}+V(x)\right]\Psi (x)=E_{\text{ges}}\cdot \Psi (x)}
∫ V d V Ψ ⋇ ( x → ) ⋅ Ψ ( x → ) = 1 {\displaystyle \int \limits _{V}^{}{\text{d}}V\Psi ^{\divideontimes }({\vec {x}})\cdot \Psi ({\vec {x}})=1}
Ψ ⋇ ( x → ) ⋅ Ψ ( x → ) {\displaystyle \Psi ^{\divideontimes }({\vec {x}})\cdot \Psi ({\vec {x}})}
p → ℏ i ∂ ∂ x {\displaystyle p\to {\frac {\mathbf {\hbar } }{\text{i}}}{\frac {\mathbf {\partial } }{\partial x}}}
E ^ k i n = − ℏ 2 2 m ∂ 2 ∂ x 2 {\displaystyle {\hat {E}}_{kin}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}}
P ( x ) = P 1 ( x ) + P 2 ( x ) {\displaystyle P(x){=}P_{1}\left(x\right)+P_{2}(x)}
p ^ x ^ Ψ ( x ) ≠ x ^ p ^ Ψ ( x ) {\displaystyle {\hat {p}}{\hat {x}}\Psi (x)\neq {\hat {x}}{\hat {p}}\Psi (x)}
⟨ A ⟩ = ∫ d x Ψ ⋇ ( x ) A ^ Ψ ( x ) {\displaystyle \left\langle A\right\rangle =\int {\text{d}}x\Psi ^{\divideontimes }(x){\hat {A}}\Psi (x)}
p ^ x ^ Ψ ( x ) = − i ℏ d d x ( x Ψ ( x ) ) . {\displaystyle {\hat {p}}{\hat {x}}\Psi (x)=-{\text{i}}\hbar {\frac {\text{d}}{{\text{d}}x}}(x\Psi (x)).}
p ^ x ^ Ψ ( x ) = − i ℏ d d x ( x Ψ ( x ) ) = i ℏ Ψ ( x ) − i ℏ x d d x Ψ ( x ) = i ℏ Ψ ( x ) + x ^ p ^ Ψ ( x ) . {\displaystyle {\begin{aligned}{\hat {p}}{\hat {x}}\Psi (x)&=-{\text{i}}\hbar {\frac {\text{d}}{{\text{d}}x}}(x\Psi (x))\\&={\text{i}}\hbar \Psi (x)-{\text{i}}\hbar x{\frac {\text{d}}{{\text{d}}x}}\Psi (x)\\&={\text{i}}\hbar \Psi (x)+{\hat {x}}{\hat {p}}\Psi (x).\\\end{aligned}}}
[ x ^ , p ^ ] Ψ ( x ) ≡ ( x ^ p ^ − p ^ x ^ ) Ψ ( x ) = i ℏ Ψ ( x ) {\displaystyle \left[{\hat {x}},{\hat {p}}\right]\Psi (x)\equiv ({\hat {x}}{\hat {p}}-{\hat {p}}{\hat {x}})\Psi (x)={\text{i}}\hbar \Psi (x)}
[ x ^ , p ^ ] = i ℏ {\displaystyle \left[{\hat {x}},{\hat {p}}\right]={\text{i}}\hbar }
[ A ^ , B ^ ] = i C ^ {\displaystyle \left[{\hat {A}},{\hat {B}}\right]={\text{i}}{\hat {C}}}
Δ A ⋅ Δ B ⩾ 1 2 | ⟨ C ^ ⟩ | ( 2 ) {\displaystyle \Delta A\cdot \Delta B\geqslant {\frac {1}{2}}\left|\left\langle {\hat {C}}\right\rangle \right\vert \qquad (2)}
Δ x ⋅ Δ p ⩾ ℏ 2 {\displaystyle \Delta x\cdot \Delta p\geqslant {\frac {\hbar }{2}}}
E ges = p 2 2 m + V ( x ) → E ^ ges = − ℏ 2 2 m ∂ 2 ∂ x 2 + V ( x ) {\displaystyle E_{\text{ges}}={\frac {p^{2}}{2m}}+V(x)\to {\hat {E}}_{\text{ges}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x)}
L z = x ⋅ p y − y ⋅ p x → L ^ z = ℏ i ( x ⋅ ∂ ∂ y − y ⋅ ∂ ∂ x ) {\displaystyle L_{z}=x\cdot p_{y}-y\cdot p_{x}\to {\hat {L}}_{z}={\frac {\hbar }{\text{i}}}\left(x\cdot {\frac {\partial }{\partial y}}-y\cdot {\frac {\partial }{\partial x}}\right)}
[ − ℏ 2 2 m d 2 d x 2 + V 0 ] Ψ ( x ) = E ges ⋅ Ψ ( x ) {\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}{\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}+V_{0}\right]\Psi (x)=E_{\text{ges}}\cdot \Psi (x)}
d 2 d x 2 Ψ ( x ) = − 2 m ℏ 2 ( E ges − V 0 ) ⋅ Ψ ( x ) {\displaystyle {\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}\Psi (x)=-{\frac {2m}{\hbar ^{2}}}(E_{\text{ges}}-V_{0})\cdot \Psi (x)}
ℏ 2 m B 2 + V 0 = E ges ( 1 ) {\displaystyle {\frac {\hbar }{2m}}B^{2}+V_{0}=E_{\text{ges}}\qquad (1)}
B = π ⋅ n a f u ¨ r n = 1 , 2 , 3 , . . . {\displaystyle B={\frac {\pi \cdot n}{a}}\qquad {\text{f}}{\ddot {\text{u}}}{\text{r}}\quad n=1,2,3,...}
E n = ℏ 2 π 2 2 m a 2 ⋅ n 2 + V 0 ( n = 1 , 2 , 3 , . . . ) {\displaystyle E_{n}={\frac {{\hbar }^{2}{\pi }^{2}}{2ma^{2}}}\cdot n^{2}+V_{0}\qquad (n=1,2,3,...)}
Ψ n ( x ) = A ⋅ sin ( n π a x ) {\displaystyle \Psi _{n}(x)=A\cdot {\text{sin}}\left({\frac {n\pi }{a}}x\right)}
∫ a 0 | Ψ ( x ) | 2 d x = 1. {\displaystyle \int _{a}^{0}\left|\Psi (x)\right\vert ^{2}{\text{d}}x=1.}
Ψ n ( x ) = 2 a sin ( n π a x ) ( n = 1 , 2 , 3 , . . . ) {\displaystyle \Psi _{n}(x)={\sqrt {\frac {2}{a}}}{\text{sin}}\left({\frac {n\pi }{a}}x\right)\quad (n=1,2,3,...)}
V ( x ) = { 0 f u ¨ r | x | > a 2 Gebiet I + III V 0 f u ¨ r | x | ≤ a 2 Gebiet II {\displaystyle V(x)={\begin{cases}0&{\mbox{f}}{\ddot {u}}{\mbox{r}}\left\vert x\right\vert >{\frac {a}{2}}\qquad {\mbox{Gebiet}}\quad {\mbox{I}}+{\mbox{III}}\\V_{0}&{\mbox{f}}{\ddot {u}}{\mbox{r}}\left\vert x\right\vert \leq {\frac {a}{2}}\qquad {\mbox{Gebiet}}\quad {\mbox{II}}\end{cases}}}
f ( n ) = { n / 2 , if n is even 3 n + 1 , if n is odd {\displaystyle f(n)={\begin{cases}n/2,&{\mbox{if }}n{\mbox{ is even}}\\3n+1,&{\mbox{if }}n{\mbox{ is odd}}\end{cases}}}
f u ¨ r | x | > a 2 ( d 2 d x 2 − γ 2 ) Ψ ( x ) = 0 mit γ 2 = 2 m ℏ 2 ( V o − E ) {\displaystyle {\mbox{f}}{\ddot {u}}{\mbox{r}}\left\vert x\right\vert >{\frac {a}{2}}\left({\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}-\gamma ^{2}\right)\Psi (x)=0\qquad {\text{mit}}\quad \gamma ^{2}={\frac {2m}{\hbar ^{2}}}(V_{o}-E)}
f u ¨ r | x | ≤ a 2 ( d 2 d x 2 − k 2 ) Ψ ( x ) = 0 mit k 2 = 2 m E ℏ 2 {\displaystyle {\mbox{f}}{\ddot {u}}{\mbox{r}}\left\vert x\right\vert \leq {\frac {a}{2}}\left({\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}-k^{2}\right)\Psi (x)=0\qquad {\text{mit}}\quad k^{2}={\frac {2mE}{\hbar ^{2}}}}
V 0 ≫ E {\displaystyle V_{0}\gg E}
γ 2 = 2 m ℏ 2 ( V o − E ) {\displaystyle \gamma ^{2}={\frac {2m}{\hbar ^{2}}}(V_{o}-E)}
( E ^ kin + V ( x ) ) Ψ ( x ) = E ⋅ Ψ ( x ) {\displaystyle \left({\hat {E}}_{\text{kin}}+V(x)\right)\Psi (x)=E\cdot \Psi (x)}
− ℏ 2 2 m d 2 d x 2 Ψ = E ⋅ Ψ {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}\Psi =E\cdot \Psi }
− ( ℏ 2 2 m d 2 d x 2 + V 0 ) Ψ = E ⋅ Ψ {\displaystyle -\left({\frac {\hbar ^{2}}{2m}}{\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}+V_{0}\right)\Psi =E\cdot \Psi }
d 2 d x 2 Ψ = − 2 m E ℏ 2 ⋅ Ψ {\displaystyle {\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}\Psi =-{\frac {2mE}{\hbar ^{2}}}\cdot \Psi }
d 2 d x 2 Ψ = − 2 m ( E − V 0 ) ℏ 2 ⋅ Ψ {\displaystyle {\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}\Psi =-{\frac {2m(E-V_{0})}{\hbar ^{2}}}\cdot \Psi }
Ψ ( x ) = { − k 2 k 1 sin ( k 1 x ) + cos ( k 1 x ) falls x ≤ 0 e x p ( − k 2 x ) falls 0 < x < a e x p ( − k 2 a ) ⋅ ( − k 2 k 1 sin ( k 1 ( x − a ) ) + cos ( k 1 ( x − a ) ) ) falls a ≤ x {\displaystyle \Psi (x)={\begin{cases}-{\frac {k_{2}}{k_{1}}}{\mbox{sin}}(k_{1}x)+{\mbox{cos}}(k_{1}x)&{\mbox{falls}}\quad x\leq 0\\exp(-k_{2}x)&{\mbox{falls}}\quad 0<x<a\\exp(-k_{2}a)\cdot \left(-{\frac {k_{2}}{k_{1}}}{\mbox{sin}}(k_{1}(x-a))+{\mbox{cos}}(k_{1}(x-a))\right)&{\mbox{falls}}\quad a\leq x\end{cases}}}
k 1 = 1 ℏ 2 m E und k 2 = 1 ℏ 2 m ( E − V 0 ) {\displaystyle k_{1}={\frac {1}{\hbar }}{\sqrt {2mE}}\quad {\text{und}}\quad k_{2}={\frac {1}{\hbar }}{\sqrt {2m(E-V_{0})}}}
T = e − 2 m ( E − V 0 ) 2 a ℏ {\displaystyle T=e^{-{\sqrt {2m(E-V_{0})}}{\frac {2a}{\hbar }}}}
1 1 H + 1 1 H → 1 2 H + 1 0 e + 0 0 v {\displaystyle {}_{1}^{1}\!{\text{H}}+{}_{1}^{1}\!{\text{H}}\to {}_{1}^{2}\!{\text{H}}+{}_{1}^{0}\!{\text{e}}+{}_{0}^{0}\!{\text{v}}}
E kin = 1 4 π ϵ 0 ⋅ e 2 1 , 0 fm {\displaystyle E_{\text{kin}}={\frac {1}{4\pi \epsilon _{0}}}\cdot {\frac {e^{2}}{1,0{\text{fm}}}}}
E ¯ kin = 3 2 k T {\displaystyle {\bar {E}}_{\text{kin}}={\frac {3}{2}}kT}
T ≈ exp ( − 2 ℏ ∫ x 0 x 2 m ( U ( r ) − U ( x ) ) d r ) {\displaystyle T\approx {\text{exp}}\left(-{\frac {2}{\hbar }}\int _{x_{0}}^{x}{\sqrt {2m(U(r)-U(x))}}{\text{d}}r\right)}
f = f Ry ( 1 m 2 − 1 n 2 ) {\displaystyle f=f_{\text{Ry}}\left({\frac {1}{m^{2}}}-{\frac {1}{n^{2}}}\right)}
h ⋅ f = E n − E m {\displaystyle h\cdot f=E_{n}-E_{m}}
E ^ kin = − ℏ 2 2 m ( d 2 d x 2 + d 2 d y 2 + d 2 d z 2 ) {\displaystyle {\hat {E}}_{\text{kin}}=-{\frac {\hbar ^{2}}{2m}}\left({\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}+{\frac {{\text{d}}^{2}}{{\text{d}}y^{2}}}+{\frac {{\text{d}}^{2}}{{\text{d}}z^{2}}}\right)}
[ − ℏ 2 2 m ( d 2 d x 2 + d 2 d y 2 + d 2 d z 2 ) + V ( x , y , z ) ] Ψ ( x , y , z ) = E ges ⋅ Ψ ( x , y , z ) {\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\left({\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}+{\frac {{\text{d}}^{2}}{{\text{d}}y^{2}}}+{\frac {{\text{d}}^{2}}{{\text{d}}z^{2}}}\right)+V(x,y,z)\right]\Psi (x,y,z)=E_{\text{ges}}\cdot \Psi (x,y,z)}
Ψ n x n y n z ( x , y , z ) = A ⋅ sin ( n x π a x ) ⋅ sin ( n y π a y ) ⋅ sin ( n z π a z ) f u ¨ r 0 ≤ x , y , z ≤ a {\displaystyle \Psi _{n_{x}n_{y}n_{z}}(x,y,z)=A\cdot {\text{sin}}\left({\frac {n_{x}\pi }{a}}x\right)\cdot {\text{sin}}\left({\frac {n_{y}\pi }{a}}y\right)\cdot {\text{sin}}\left({\frac {n_{z}\pi }{a}}z\right)\qquad {\text{f}}{\ddot {u}}{\text{r}}\quad 0\leq x,y,z\leq a}
[ − ℏ 2 2 m ( n x 2 π 2 a 2 − n y 2 π 2 a 2 − n z 2 π 2 a 2 ) ] Ψ n x n y n z ( x , y , z ) = E ges ⋅ Ψ n x n y n z ( x , y , z ) {\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\left({\frac {n_{x}^{2}\pi ^{2}}{a^{2}}}-{\frac {n_{y}^{2}\pi ^{2}}{a^{2}}}-{\frac {n_{z}^{2}\pi ^{2}}{a^{2}}}\right)\right]\Psi _{n_{x}n_{y}n_{z}}(x,y,z)=E_{\text{ges}}\cdot \Psi _{n_{x}n_{y}n_{z}}(x,y,z)}
E ges = ℏ 2 π 2 2 m a 2 ( n x 2 + n y 2 + n z 2 ) + V 0 mit ( n x , n y , n z = 1 , 2 , 3 , . . . ) {\displaystyle E_{\text{ges}}={\frac {\hbar ^{2}\pi ^{2}}{2ma^{2}}}(n_{x}^{2}+n_{y}^{2}+n_{z}^{2})+V_{0}\qquad {\text{mit}}\quad (n_{x},n_{y},n_{z}=1,2,3,...)}
R = − e 2 4 π ϵ 0 E ges {\displaystyle R=-{\frac {e^{2}}{4\pi \epsilon _{0}E_{\text{ges}}}}}
W 0 = W ( r = 1 2 R ) = W ( r = − 1 2 e 2 4 π ϵ 0 E ) {\displaystyle W_{0}=W\left(r={\frac {1}{2}}R\right)=W\left(r=-{\frac {1}{2}}{\frac {e^{2}}{4\pi \epsilon _{0}E}}\right)}
W = 2 ⋅ E ges = − 2 ⋅ | E ges | {\displaystyle W=2\cdot E_{\text{ges}}=-2\cdot \left\vert E_{\text{ges}}\right\vert }
W 0 = 2 ⋅ E ges {\displaystyle W_{0}=2\cdot E_{\text{ges}}}
R = e 2 4 π ϵ 0 | E ges | ( 1 ) {\displaystyle R={\frac {e^{2}}{4\pi \epsilon _{0}\left\vert E_{\text{ges}}\right\vert }}\qquad (1)}
E ges = ℏ 2 π 2 2 m ( 2 R ) 2 ⋅ 3 n 2 + W 0 ( 3 ) {\displaystyle E_{\text{ges}}={\frac {\hbar ^{2}\pi ^{2}}{2m(2R)^{2}}}\cdot 3n^{2}+W_{0}\qquad (3)}
E ges = ℏ 2 π 2 2 m ( 2 R ) 2 ⋅ ( n x 2 + n y 2 + n z 2 ) + W 0 ( 2 ) {\displaystyle E_{\text{ges}}={\frac {\hbar ^{2}\pi ^{2}}{2m(2R)^{2}}}\cdot (n_{x}^{2}+n_{y}^{2}+n_{z}^{2})+W_{0}\qquad (2)}
E ges = ℏ 2 π 2 2 m ⋅ 3 n 2 ⋅ ( 4 π ϵ 0 ) 2 E ges 2 4 e 4 + 2 E ges {\displaystyle E_{\text{ges}}={\frac {\hbar ^{2}\pi ^{2}}{2m}}\cdot 3n^{2}\cdot {\frac {(4\pi \epsilon _{0})^{2}E_{\text{ges}}^{2}}{4e^{4}}}+2E_{\text{ges}}}
E ges = − 16 3 π 2 ⋅ m e 4 2 ℏ 2 ( 4 π ϵ 0 ) 2 ⋅ 1 n 2 {\displaystyle E_{\text{ges}}=-{\frac {16}{3\pi ^{2}}}\cdot {\frac {me^{4}}{2\hbar ^{2}(4\pi \epsilon _{0})^{2}}}\cdot {\frac {1}{n^{2}}}}
W 0 = e 2 4 π ϵ 0 ⋅ 2 ⋅ 4 π ϵ 0 E e 2 {\displaystyle W_{0}={\frac {e^{2}}{4\pi \epsilon _{0}}}\cdot 2\cdot {\frac {4\pi \epsilon _{0}E}{e^{2}}}}
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![{\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}{\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}+V_{0}\right]\Psi (x)=E_{\text{ges}}\cdot \Psi (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1be343ef7857960b1d6ef955f97b32ad9cddf276)
![{\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\left({\frac {{\text{d}}^{2}}{{\text{d}}x^{2}}}+{\frac {{\text{d}}^{2}}{{\text{d}}y^{2}}}+{\frac {{\text{d}}^{2}}{{\text{d}}z^{2}}}\right)+V(x,y,z)\right]\Psi (x,y,z)=E_{\text{ges}}\cdot \Psi (x,y,z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4cd991186276fe6426955c92411a6709416c4e)
![{\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}\left({\frac {n_{x}^{2}\pi ^{2}}{a^{2}}}-{\frac {n_{y}^{2}\pi ^{2}}{a^{2}}}-{\frac {n_{z}^{2}\pi ^{2}}{a^{2}}}\right)\right]\Psi _{n_{x}n_{y}n_{z}}(x,y,z)=E_{\text{ges}}\cdot \Psi _{n_{x}n_{y}n_{z}}(x,y,z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a226b7c1d5349f2c49ab3282158ef365cb246692)
