ਫਰਮਾ:ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ ਓਪਰੇਟਰ

ਓਪਰੇਟਰ (ਸਾਂਝਾ ਨਾਮ)	ਕਾਰਟੀਜ਼ੀਅਨ ਕੰਪੋਨੈਂਟ	ਆਮ ਪਰਿਭਾਸ਼ਾ	SI ਯੂਨਿਟ	ਡਾਇਮੈਂਸ਼ਨ
ਪੁਜੀਸ਼ਨ	${\begin{aligned}{\hat {x}}=x\\{\hat {y}}=y\\{\hat {z}}=z\end{aligned}}$	$\mathbf {\hat {r}} =\mathbf {r} \,\!$	m	[L]
ਮੋਮੈਂਟਮ	ਆਮ ${\begin{aligned}{\hat {p}}_{x}&=-i\hbar {\frac {\partial }{\partial x}}\\{\hat {p}}_{y}&=-i\hbar {\frac {\partial }{\partial y}}\\{\hat {p}}_{z}&=-i\hbar {\frac {\partial }{\partial z}}\end{aligned}}$	ਆਮ $\mathbf {\hat {p}} =-i\hbar \nabla \,\!$	J s m−1 = N s	[M] [L] [T]−1
	ਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ ${\begin{aligned}{\hat {p}}_{x}=-i\hbar {\frac {\partial }{\partial x}}-qA_{x}\\{\hat {p}}_{y}=-i\hbar {\frac {\partial }{\partial y}}-qA_{y}\\{\hat {p}}_{z}=-i\hbar {\frac {\partial }{\partial z}}-qA_{z}\end{aligned}}$	ਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ (ਕਾਇਨੈਟਿਕ ਮੋਮੈਂਟਮ ਵਰਤਦਾ ਹੈ, A = ਵੈਕਟਰ ਪੁਟੇਂਸ਼ਲ) ${\begin{aligned}\mathbf {\hat {p}} &=\mathbf {\hat {P}} -q\mathbf {A} \\&=-i\hbar \nabla -q\mathbf {A} \\\end{aligned}}\,\!$	J s m−1 = N s	[M] [L] [T]−1
ਕਾਇਨੈਟਿਕ ਐਨਰਜੀ	ਟ੍ਰਾਂਸਲੇਸ਼ਨ ${\begin{aligned}{\hat {T}}_{x}&=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\\{\hat {T}}_{y}&=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial y^{2}}}\\{\hat {T}}_{z}&=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}\\\end{aligned}}$	${\begin{aligned}{\hat {T}}&={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}\\&={\frac {(-i\hbar \nabla )\cdot (-i\hbar \nabla )}{2m}}\\&={\frac {-\hbar ^{2}}{2m}}\nabla ^{2}\end{aligned}}\,\!$	J	[M] [L]2 [T]−2
	ਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ ${\begin{aligned}{\hat {T}}_{x}&={\frac {1}{2m}}\left(-i\hbar {\frac {\partial }{\partial x}}-qA_{x}\right)^{2}\\{\hat {T}}_{y}&={\frac {1}{2m}}\left(-i\hbar {\frac {\partial }{\partial y}}-qA_{y}\right)^{2}\\{\hat {T}}_{z}&={\frac {1}{2m}}\left(-i\hbar {\frac {\partial }{\partial z}}-qA_{z}\right)^{2}\end{aligned}}\,\!$	ਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ (A = ਵੈਕਟਰ ਪੁਟੈਂਸ਼ਲ) ${\begin{aligned}{\hat {T}}&={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}\\&={\frac {1}{2m}}(-i\hbar \nabla -q\mathbf {A} )\cdot (-i\hbar \nabla -q\mathbf {A} )\\&={\frac {1}{2m}}(-i\hbar \nabla -q\mathbf {A} )^{2}\end{aligned}}\,\!$	J	[M] [L]2 [T]−2
	ਰੋਟੇਸ਼ਨ (I = ਇਨ੍ਰਸ਼ੀਆ ਦੀ ਮੋਮੈਂਟ) ${\begin{aligned}{\hat {T}}_{xx}&={\frac {{\hat {J}}_{x}^{2}}{2I_{xx}}}\\{\hat {T}}_{yy}&={\frac {{\hat {J}}_{y}^{2}}{2I_{yy}}}\\{\hat {T}}_{zz}&={\frac {{\hat {J}}_{z}^{2}}{2I_{zz}}}\\\end{aligned}}\,\!$	ਰੋਟੇਸ਼ਨ ${\hat {T}}={\frac {\mathbf {\hat {J}} \cdot \mathbf {\hat {J}} }{2I}}\,\!$	J	[M] [L]2 [T]−2
ਪੁਟੇਂਸ਼ਲ ਐਨਰਜੀ	N/A	${\hat {V}}=V\left(\mathbf {r} ,t\right)=V\,\!$	J	[M] [L]2 [T]−2
ਕੁੱਲ ਐਨਰਜੀ	N/A	ਸਮੇਂ-ਤੇ-ਨਿਰਭਰ ਪੁਟੈਂਸ਼ਲ: ${\hat {E}}=i\hbar {\frac {\partial }{\partial t}}\,\!$ ਸਮੇਂ-ਤੇ-ਨਿਰਭਰ: ${\hat {E}}=E\,\!$	J	[M] [L]2 [T]−2
ਹੈਮਿਲਟੋਨੀਅਨ		${\begin{aligned}{\hat {H}}&={\hat {T}}+{\hat {V}}\\&={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}+V\\&={\frac {{\hat {p}}^{2}}{2m}}+V\\\end{aligned}}\,\!$	J	[M] [L]2 [T]−2
ਐਂਗੁਲਰ ਮੋਮੈਂਟਮ ਓਪਰੇਟਰ	${\begin{aligned}{\hat {L}}_{x}&=-i\hbar \left(y{\partial \over \partial z}-z{\partial \over \partial y}\right)\\{\hat {L}}_{y}&=-i\hbar \left(z{\partial \over \partial x}-x{\partial \over \partial z}\right)\\{\hat {L}}_{z}&=-i\hbar \left(x{\partial \over \partial y}-y{\partial \over \partial x}\right)\end{aligned}}$	$\mathbf {\hat {L}} =\mathbf {r} \times -i\hbar \nabla$	J s = N s m−1	[M] [L]2 [T]−1
ਸਪਿੱਨ ਐਂਗੁਲਰ ਮੋਮੈਂਟਮ	${\begin{aligned}{\hat {S}}_{x}={\hbar \over 2}\sigma _{x}\\{\hat {S}}_{y}={\hbar \over 2}\sigma _{y}\\{\hat {S}}_{z}={\hbar \over 2}\sigma _{z}\end{aligned}}$ ਜਿੱਥੇ $\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$ $\sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}$ $\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$ ਸਪਿੱਨ-½ ਕਣਾਂ ਵਾਸਤੇ ਪੌਲੀ ਮੈਟ੍ਰਿਕਸ ਹਨ।	$\mathbf {\hat {S}} ={\hbar \over 2}{\boldsymbol {\sigma }}\,\!$ ਜਿੱਥੇ σ ਪੌਲੀ ਮੈਟ੍ਰਿਕਸਾਂ ਨਾਮਕ ਕੰਪੋਨੈਂਟਾਂ ਦਾ ਵੈਕਟਰ ਹੁੰਦਾ ਹੈ।	J s = N s m−1	[M] [L]2 [T]−1
ਕੁੱਲ ਐਂਗੁਲਰ ਮੋਮੈਂਟਮ	${\begin{aligned}{\hat {J}}_{x}&={\hat {L}}_{x}+{\hat {S}}_{x}\\{\hat {J}}_{y}&={\hat {L}}_{y}+{\hat {S}}_{y}\\{\hat {J}}_{z}&={\hat {L}}_{z}+{\hat {S}}_{z}\end{aligned}}$	${\begin{aligned}\mathbf {\hat {J}} &=\mathbf {\hat {L}} +\mathbf {\hat {S}} \\&=-i\hbar \mathbf {r} \times \nabla +{\frac {\hbar }{2}}{\boldsymbol {\sigma }}\end{aligned}}$	J s = N s m−1	[M] [L]2 [T]−1
ਟ੍ਰਾਂਜ਼ੀਸ਼ਨ ਡਾਈਪੋਲ ਮੋਮੈਂਟ (ਇਲੈਕਟ੍ਰਿਕ)	${\begin{aligned}{\hat {d}}_{x}&=q{\hat {x}}\\{\hat {d}}_{y}&=q{\hat {y}}\\{\hat {d}}_{z}&=q{\hat {z}}\end{aligned}}$	$\mathbf {\hat {d}} =q\mathbf {\hat {r}}$	C m	[I] [T] [L]