- 自然科学(数学・物理学・電磁気学・力学・熱力学・熱化学・化学)
関数
についての微分の定義
関数
についての偏微分の定義。
![{\displaystyle {\tfrac {\partial f}{\partial x}}=f_{x}(x,y)=\left({\tfrac {\partial f}{\partial x}}\right)_{y}\equiv \lim _{\Delta {x}\to {0}}{\tfrac {f(x+\Delta {x},y)-f(x,y)}{\Delta {x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7011323d07633b06f7244edf9e45bd3762f7ab51)
![{\displaystyle {\tfrac {\partial f}{\partial y}}=f_{y}(x,y)=\left({\tfrac {\partial f}{\partial y}}\right)_{x}\equiv \lim _{\Delta {y}\to {0}}{\tfrac {f(x,y+\Delta {y})-f(x,y)}{\Delta {y}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feeaa21e17d038f93b4132a43bef36fe7108a1f7)
ニュートン法
3次元ベクトルに関するベクトル解析の演算子及び作用素。
- ナブラ
![{\displaystyle \nabla \equiv {\tfrac {\partial }{\partial x}}\,\mathbf {i} +{\tfrac {\partial }{\partial y}}\,\mathbf {j} +{\tfrac {\partial }{\partial z}}\,\mathbf {k} =\left({\tfrac {\partial }{\partial {x}}},{\tfrac {\partial }{\partial {y}}},{\tfrac {\partial }{\partial {z}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89995b44c528144f5b74d36c4b75c2b4fb29512e)
- 勾配
![{\displaystyle {\mathsf {grad}}\,\phi =\nabla \phi ={\tfrac {\partial \phi }{\partial x}}\,\mathbf {i} +{\tfrac {\partial \phi }{\partial y}}\,\mathbf {j} +{\tfrac {\partial \phi }{\partial z}}\,\mathbf {k} =\left({\tfrac {\partial \phi }{\partial {x}}},{\tfrac {\partial \phi }{\partial {y}}},{\tfrac {\partial \phi }{\partial {z}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4a94521ddcf69a8b6d8fa1cf96f4bdcf61efdf)
- 発散
![{\displaystyle {\mathsf {div}}\,{\boldsymbol {X}}=\nabla \cdot {\boldsymbol {X}}={\tfrac {\partial {X_{x}}}{\partial {x}}}+{\tfrac {\partial {X_{y}}}{\partial {y}}}+{\tfrac {\partial {X_{z}}}{\partial {z}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8756391a7361a364ea8e9375056c1dcc89f98863)
- 回転
![{\displaystyle {\begin{aligned}{\mathsf {rot}}\,{\boldsymbol {E}}&=\nabla \times {\boldsymbol {E}}={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\{\tfrac {\partial }{\partial x}}&{\tfrac {\partial }{\partial y}}&{\tfrac {\partial }{\partial z}}\\E_{x}&E_{y}&E_{z}\end{vmatrix}}\\&=\left({\tfrac {\partial E_{z}}{\partial y}}-{\tfrac {\partial E_{y}}{\partial z}}\right)\mathbf {i} +\left({\tfrac {\partial E_{x}}{\partial z}}-{\tfrac {\partial E_{z}}{\partial x}}\right)\mathbf {j} +\left({\tfrac {\partial E_{y}}{\partial x}}-{\tfrac {\partial E_{x}}{\partial y}}\right)\mathbf {k} \\&=\left({\tfrac {\partial {E_{z}}}{\partial {y}}}-{\tfrac {\partial {E_{y}}}{\partial {z}}},{\tfrac {\partial {E_{x}}}{\partial {z}}}-{\tfrac {\partial {E_{z}}}{\partial {x}}},{\tfrac {\partial {E_{y}}}{\partial {x}}}-{\tfrac {\partial {E_{x}}}{\partial {y}}}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/853461b26efc5c0e8d3fbf7571eddbfe269d29ec)
- ラプラシアン
![{\displaystyle {\begin{aligned}\Delta \,\phi &=\nabla ^{2}\,\phi =\nabla \cdot \nabla \,\phi ={\mathsf {div}}\,{\mathsf {grad}}\,\phi \\&={\tfrac {\partial }{\partial {x}}}\left({\tfrac {\partial \phi }{\partial {x}}}\right)+{\tfrac {\partial }{\partial {y}}}\left({\tfrac {\partial \phi }{\partial {y}}}\right)+{\tfrac {\partial }{\partial {z}}}\left({\tfrac {\partial \phi }{\partial {z}}}\right)={\tfrac {\partial ^{2}\phi }{\partial {x^{2}}}}+{\tfrac {\partial ^{2}\phi }{\partial {y^{2}}}}+{\tfrac {\partial ^{2}\phi }{\partial {z^{2}}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5383b44f05afd0c5979c35718d96adaef5100bfe)
- ダランベルシアン
![{\displaystyle \Box \equiv \Delta -{\tfrac {1}{c^{2}}}{\tfrac {\partial ^{2}}{\partial t^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/942e27b9e7ba864418686ce95df6b8a61f1559a5)
![{\displaystyle {\begin{aligned}\nabla \times \nabla \times {\boldsymbol {A}}&={\mathsf {rot}}({\mathsf {rot}}\,{\boldsymbol {A}})=\nabla (\nabla \cdot {\boldsymbol {A}})-\nabla ^{2}\,{\boldsymbol {A}}\\&=\nabla \,{\mathsf {div}}\,{\boldsymbol {A}}-\nabla \cdot \nabla \,{\boldsymbol {A}}={\mathsf {grad}}\,{\mathsf {div}}\,{\boldsymbol {A}}-\Delta \,{\boldsymbol {A}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/261bf9f55faff8dfc83fc1d1198eca66a48f5e28)
関数の定積分
閉曲線Cの線積分
重積分。
- 平面Dの二重積分
![{\displaystyle \iint _{D}{f(x,y)}\,dx\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2694eb17fa395c8292525ab92e788f5f3e11569a)
- 立体の三重積分
![{\displaystyle \iiint \limits _{\text{cuboid}}{f(x,y,z)}\,dx\,dy\,dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c18e6663dac6620f36effc6c2364d99ff3012e1)
フックの法則。
![{\displaystyle F=-kx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aace79ea36db4ebc9f83a00c4198f4c054f5b4fb)
![{\displaystyle U={\tfrac {1}{2}}kx^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63fb6c25dcf366cd95ed3c203a2e0ef55876a235)
宇宙の温度
ハッブルの法則。
![{\displaystyle {\boldsymbol {v}}=H_{0}{\boldsymbol {r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abe5517f357a185f8f9d118cfe3c052056759c1f)
![{\displaystyle H_{0}\equiv {\tfrac {\dot {a}}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c97de34492c4fc66fd9bff2a9d9dac17edf8a9d)
オイラーの公式
より導出される世界で最も美しい数式の一つとされるオイラーの等式
虚数
、即ち
。
熱力学第一法則
ポアソンの関係式。
![{\displaystyle PV^{\gamma }=k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e446c2bfdde45db18995d45e7cc8965ed11bc1ec)
![{\displaystyle TV^{\gamma -1}=k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8ae43f053cd953d66465266e1a77aebfc41ad8)
マイヤーの関係式
ファン・デル・ワールスの状態方程式
ミカエリス・メンテン式
ラインウィーバー=バークプロット
ランベルト・ベールの法則
吸光度
光電効果の公式
マクスウェル=ボルツマン分布
ボース=アインシュタイン分布
フェルミ=ディラック分布
黒体及び熱放射に関する法則。
- ウィーンの変位則
![{\displaystyle \lambda _{\text{max}}={\tfrac {b}{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30935c6c4359ebf48051615ecaa256f8bb38ba37)
![{\displaystyle R(\lambda ,T)={\tfrac {c_{1}}{\lambda ^{5}}}e^{-{\tfrac {c_{2}}{\lambda T}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a8b3949fabd84fd247f75cda27b9763f85b7b5)
![{\displaystyle R(\nu ,T)={\tfrac {c_{1}\nu ^{3}}{c^{4}}}e^{-{\tfrac {c_{2}\nu }{cT}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1225604a7406eb7e77ac61301dc081b178ccab)
![{\displaystyle I(\lambda ,T)={\tfrac {1}{\pi }}R(\lambda ,T)={\tfrac {c_{1L}}{\lambda ^{5}}}e^{-{\tfrac {c_{2}}{\lambda T}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10a16412110356a330edee3f518261b763bbd251)
![{\displaystyle I(\nu ,T)={\tfrac {c_{1L}\nu ^{3}}{c^{4}}}e^{-{\tfrac {c_{2}\nu }{cT}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12da92eb74fad7417f57ee1cedcf10a3e584309c)
![{\displaystyle c_{1}=2\pi hc^{2},~c_{1L}=2hc^{2},~c_{2}={\tfrac {hc}{k_{\text{B}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350c0806292e54cf409b7ef6b17135bf288707ab)
![{\displaystyle u(\nu ,T)={\tfrac {8\pi h\nu ^{3}}{c^{3}}}e^{-{\tfrac {h\nu }{k_{\text{B}}T}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fde9e66cbf495ddde32164d44007866a4abf7878)
![{\displaystyle u(\lambda ,T)={\tfrac {8\pi k_{\text{B}}T}{\lambda ^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb88877f9654deb4c7e1b02d7a11c8b3026a039)
![{\displaystyle u(\nu ,T)={\tfrac {8\pi \nu ^{2}k_{\text{B}}T}{c^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30e2ce3a24f8e36a619fdbcd057991d81bfb4d7f)
![{\displaystyle I=\sigma T^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77f3a1a6117a249f1c7b5a751bdfc4f675b25aee)
![{\displaystyle \sigma ={\tfrac {2\pi ^{5}k_{\mathrm {B} }^{4}}{15c^{2}h^{3}}}={\tfrac {\pi ^{2}k_{\mathrm {B} }^{4}}{60c^{2}\hbar ^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2f0944e7d573d33f4e55aaae98418b183d128f)
![{\displaystyle I(\nu ,T)={\tfrac {2h\nu ^{3}}{c^{2}}}{\tfrac {1}{e^{h\nu /k_{\mathrm {B} }T}-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b310ded9cc7d61e059a598477aeb572148f76e2)
![{\displaystyle I'(\lambda ,T)={\tfrac {2hc^{2}}{\lambda ^{5}}}{\tfrac {1}{e^{hc/\lambda k_{\mathrm {B} }T}-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/807e6489be517f21bf9e707a3837452a49f623e1)
![{\displaystyle u(\nu ,T)={\tfrac {4\pi }{c}}I(\nu ,T)={\tfrac {8\pi h\nu ^{3}}{c^{3}}}{\tfrac {1}{e^{h\nu /k_{\mathrm {B} }T}-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/424dfb5737c7e03eac3287542cd9160267bb9f11)
![{\displaystyle u'(\lambda ,T)={\tfrac {4\pi }{c}}I'(\lambda ,T)={\tfrac {8\pi hc}{\lambda ^{5}}}{\tfrac {1}{e^{hc/\lambda k_{\mathrm {B} }T}-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f364890d84ecfdb57e44c657b545c0c0b7e8c14)
部分モル体積。
![{\displaystyle {\begin{aligned}V&=n_{1}\left({\tfrac {\partial V}{\partial n_{1}}}\right)_{n_{2},T,P}+n_{2}\left({\tfrac {\partial V}{\partial n_{2}}}\right)_{n_{1},T,P}\\&=n_{1}{\bar {V_{1}}}+n_{2}{\bar {V_{2}}}=n_{1}V_{1}^{*}+n_{2}\phi _{V}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e3954b4af16a86d9fc3d38afde519fc0aa44ed)
![{\displaystyle dV=\left({\tfrac {\partial V}{\partial n_{1}}}\right)_{n_{2},T,P}dn_{1}+\left({\tfrac {\partial V}{\partial n_{2}}}\right)_{n_{1},T,P}dn_{2}={\bar {V_{1}}}dn_{1}+{\bar {V_{2}}}dn_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7552992dce6a572dc5f2cd13e9edd2e27eb81e8f)
![{\displaystyle \phi _{V}=\phi _{V}^{\circ }+S_{V}{\sqrt {m}}={\tfrac {V-n_{1}V_{1}^{*}}{n_{2}}}={\tfrac {1}{m}}\left[{\tfrac {1000+mM}{d}}-{\tfrac {1000}{d_{0}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef6d1fc3a997e42efd774786849b86eeb7b0cee3)
![{\displaystyle {\begin{aligned}{\bar {V_{2}}}&=\left({\tfrac {\partial V}{\partial n_{2}}}\right)_{n_{1},T,P}=\phi _{V}+n_{2}\left({\tfrac {\partial V}{\partial n_{2}}}\right)_{n_{1},T,P}\\&=\phi _{V}+{\tfrac {\sqrt {m}}{2}}\left({\tfrac {\partial \phi _{V}}{\partial {\sqrt {m}}}}\right)=\phi _{V}+{\tfrac {\sqrt {m}}{2}}S_{V}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/510ec5ff8f23434655ce1bdbaab7a5451f79428f)
光合成
フーリエの法則
拡散方程式
物性論におけるアインシュタインの比熱式
ヴィーデマン=フランツ則。
![{\displaystyle {\tfrac {\mathrm {K} }{\sigma }}=LT}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9216fd1badf2979f6a8d9ceb9bc685a3ee16e146)
![{\displaystyle L={\tfrac {\mathrm {K} }{\sigma T}}={\tfrac {(\pi k_{\text{B}})^{2}}{3e^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29db0f62dbef11c0aa2356695b389bb404a53156)
磁気や電磁気に関する法則。
- 磁場
![{\displaystyle \mathbf {H} =-{\tfrac {1}{4\pi \mu _{0}}}\,{\mathsf {grad}}\,{\tfrac {\mathbf {m} \cdot \mathbf {r} }{r^{3}}}=-{\tfrac {1}{4\pi \mu _{0}}}\,\nabla \,{\tfrac {\mathbf {m} \cdot \mathbf {r} }{r^{3}}}=-{\mathsf {grad}}\,\phi _{m}=-\nabla \,\phi _{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2420aa81cecc182c0f1029831ce69fb5705059df)
- ポテンシャル
![{\displaystyle \phi _{m}={\tfrac {1}{4\pi \mu _{0}}}{\tfrac {\mathbf {m} \cdot \mathbf {r} }{r^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c165026d9df2c5270a2c62bfb11736f23c77016)
- 磁束密度
![{\displaystyle \mathbf {B} =\mu _{0}\mathbf {H} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d2fd45a5e63db4904cb79193388449e8cb2ccf8)
- 磁化
![{\displaystyle \mathbf {M} =\chi _{m}\mathbf {B} =\chi _{m}\mu _{0}{H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69461f13e1416cf46c9d749edd32805ca68d4599)
- アンペールの力
![{\displaystyle \mathbf {F} =I\mathbf {s} \times \mathbf {B} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0abe907393f3fea02d528d4b48e1651813ea6bcc)
- ローレンツ力
![{\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc7325732c56e5caeacd2005cd803c9ed5e9d8d)
- ビオ・サバールの法則
![{\displaystyle d\mathbf {B} ={\tfrac {\mu _{0}}{4\pi }}{\tfrac {Id\mathbf {s} \times \mathbf {r} }{r^{3}}}={\tfrac {\mu _{0}}{4\pi }}{\tfrac {I\sin \theta }{r^{2}}}\,d\mathbf {s} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/753c79a80b0314005baab2116794359ecefc8832)
- クーロン力
![{\displaystyle \mathbf {F} ={\tfrac {1}{4\pi \mu _{0}}}{\tfrac {q_{m1}q_{m2}}{r^{2}}}\mathbf {\hat {r}} =q_{m}\mathbf {H} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/55237bd0c028263d4709be6bfbc3722db4f69571)
- 磁場
![{\displaystyle \mathbf {H} (r)={\tfrac {1}{4\pi \mu _{0}}}{\tfrac {q_{m}}{r^{2}}}\mathbf {\hat {r}} =-{\tfrac {1}{4\pi \mu _{0}}}\,{\mathsf {grad}}\,{\tfrac {q_{m}}{r}}=-{\tfrac {1}{4\pi \mu _{0}}}\nabla \,{\tfrac {q_{m}}{r}}=-{\mathsf {grad}}\,\phi _{m}=-\nabla \,\phi _{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5edcf43faabbacc6b82644c01fb4ea603796321)
- 磁場ポテンシャル
![{\displaystyle \phi _{m}(r)={\tfrac {1}{4\pi \mu _{0}}}{\tfrac {q_{m}}{r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2134db1346cf4b4d798d22e43f91d3a550612fb6)
![{\displaystyle \mathbf {\hat {r}} ={\tfrac {\mathbf {r} }{|\mathbf {r} |}}={\tfrac {\mathbf {r} }{r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64c11d76bc7a49ff39eebfdc53c4d9d27958bdb8)
![{\displaystyle \mathbf {m} =q_{m}\mathbf {r} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/49761be373e07c4438682cd170750a1ca9999f2f)
![{\displaystyle m=\mu _{0}IS}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e06e8293075220bf3232de650353d92f8dac456)
- ポテンシャル
![{\displaystyle U_{m}=-{\tfrac {1}{4\pi \mu _{0}}}\left[{\tfrac {3(\mathbf {m_{1}} \cdot \mathbf {r} )(\mathbf {m_{2}} \cdot \mathbf {r} )}{r^{5}}}-{\tfrac {\mathbf {m_{1}} \cdot \mathbf {m_{2}} }{r^{3}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39ee5bf1913a893c1a26863bef124fbf35808e40)
![{\displaystyle \mathbf {M} =C{\tfrac {\mathbf {B} }{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f692b136e9e9a04ee67a1810afe6b6975ae37f6)
![{\displaystyle \chi _{m}={\tfrac {C}{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a77c404b3f130bec6f6af5ef9c7c7a38c4c135a)
![{\displaystyle \mathbf {M} =C{\tfrac {\mathbf {B} }{T-\theta _{p}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c59afd34e8b3370e0c568d749f329f9ef79615e0)
![{\displaystyle \chi _{m}={\tfrac {C}{T-\theta _{p}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4541cfee9c92538808b3db74dfee29aa7a48729d)
デバイの長さ。
- プラズマの場合
![{\displaystyle \lambda _{D}={\sqrt {\tfrac {\varepsilon _{0}k_{\mathrm {B} }T}{n_{\mathrm {e} }e^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae36bb8c7542ec7bc6f0385bebcc3ac67888d7d)
- 強電解質の場合
![{\displaystyle \varkappa ^{-1}={\sqrt {\tfrac {\varepsilon k_{\mathrm {B} }T}{2N_{\mathrm {A} }e^{2}I}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38f049c79e3d91c3c802df38ee73017c02c3a9fe)
プラズマ振動数
波動方程式。
![{\displaystyle {\tfrac {1}{v^{2}}}{\tfrac {\partial ^{2}u}{\partial t^{2}}}={\tfrac {\partial ^{2}u}{\partial x^{2}}}+{\tfrac {\partial ^{2}u}{\partial y^{2}}}+{\tfrac {\partial ^{2}u}{\partial z^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26637c880b40ea7ee994f3bc6993558349cc4a3)
![{\displaystyle u=u(x,y,z,t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/447d8aa4ecb76098b36ef8f51d1e1852364eb16d)
![{\displaystyle \Delta \,u={\tfrac {1}{v^{2}}}{\tfrac {\partial ^{2}u}{\partial t^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63289abac47bc68ea44d85217b64861993282b24)
![{\displaystyle \square \,u=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ff8305e90f4068386a5e93d6287f8ea3a4d815)
![{\displaystyle v^{2}\,\Delta \,\psi ={\tfrac {\partial ^{2}\psi }{\partial t^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4353a31c08b8d7d15b5490afc59314359289ccb0)
シュレーディンガー方程式。
- ハミルトニアン
![{\displaystyle {\hat {H}}=-{\tfrac {\hbar ^{2}}{2m}}\Delta +V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e6778f379d7fd5cced49720817f8d43cd3cb89)
![{\displaystyle {\hat {H}}\psi (t)=i\hbar \left({\tfrac {\partial \psi }{\partial t}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e20055ce4a7c957447a9cb1a3a645b663d2713f)
![{\displaystyle {\tfrac {\partial ^{2}\psi }{\partial x^{2}}}+{\tfrac {\partial ^{2}\psi }{\partial y^{2}}}+{\tfrac {\partial ^{2}\psi }{\partial z^{2}}}+{\tfrac {\partial ^{2}\psi }{\partial t^{2}}}+{\tfrac {4\pi mi}{h}}\left({\tfrac {\partial \psi }{\partial t}}\right)-{\tfrac {8\pi ^{2}m}{h^{2}}}V(x,y,z,t)\psi (x,y,z,t)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a70e8bde57b694fff9ba26f26f9aabf395bfc07d)
![{\displaystyle {\hat {H}}\psi (x,y,z)=E\psi (x,y,z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0f7aa8f41e0a0410b84e61b7323332216a5230)
![{\displaystyle {\tfrac {\partial ^{2}\psi }{\partial {x}^{2}}}+{\tfrac {\partial ^{2}\psi }{\partial {y}^{2}}}+{\tfrac {\partial ^{2}\psi }{\partial {z}^{2}}}+{\tfrac {8\pi ^{2}m}{h^{2}}}\left[E-V(x,y,z)\right]\psi (x,y,z)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46f712230497b5f964f8876e3226a6a3658aa7c9)