{"id":319,"date":"2026-08-12T07:52:09","date_gmt":"2026-08-12T04:22:09","guid":{"rendered":"https:\/\/sciencio.ir\/?p=319"},"modified":"2026-08-12T07:52:09","modified_gmt":"2026-08-12T04:22:09","slug":"9-%d8%b9%d8%af%d9%85-%d9%82%d8%b7%d8%b9%db%8c%d8%aa","status":"publish","type":"post","link":"https:\/\/sciencio.ir\/?p=319","title":{"rendered":"9 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a"},"content":{"rendered":"

\u0627\u0632 \u06a9\u0627\u0631\u0628\u0631\u062f\u0647\u0627\u06cc \u062c\u0628\u0631 \u062c\u0627\u0628\u062c\u0627\u06af\u0631\u060c \u0645\u062d\u0627\u0633\u0628\u0647 \u0631\u0627\u0628\u0637\u0647 \u06a9\u0644\u06cc \u0628\u0631\u0627\u06cc \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0647\u0627\u06cc \u062f\u0648 \u0639\u0645\u0644\u06af\u0631 \u062f\u0644\u062e\u0648\u0627\u0647 \u0627\u0633\u062a. \u0631\u0627\u0628\u0637\u0647 \u0627\u06cc \u06a9\u0647 \u0631\u0648\u0627\u0628\u0637 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0647\u0627\u06cc\u0632\u0646\u0628\u0631\u06af \u0631\u0627 \u0646\u062a\u06cc\u062c\u0647 \u0645\u06cc \u062f\u0647\u062f.<\/p>\n

1-9 \u062a\u0639\u0631\u06cc\u0641<\/h3>\n

\u0645\u0642\u062f\u0627\u0631 \u0686\u0634\u0645\u062f\u0627\u0634\u062a\u06cc \u0639\u0645\u0644\u06af\u0631\u0647\u0627\u06cc \u0647\u0631\u0645\u06cc\u062a\u06cc $\\hat{A}$ \u0648 $\\hat{B}$\u060c \u0646\u0633\u0628\u062a \u0628\u0647 \u0628\u0631\u062f\u0627\u0631 \u062d\u0627\u0644\u062a \u0628\u0647\u0646\u062c\u0627\u0631\u00a0<\/strong>$| \\psi \\rangle $ \u0628\u0647 \u0635\u0648\u0631\u062a \u0632\u06cc\u0631 \u0627\u0633\u062a:<\/p>\n

(1)<\/strong><\/p>\n

$\\large \\langle \\hat{A} \\rangle=\\langle \\psi |\\hat{A}| \\psi \\rangle \\; , \\; \\langle \\hat{B} \\rangle=\\langle \\psi |\\hat{B}| \\psi \\rangle$<\/p>\n

\u0639\u0645\u0644\u06af\u0631\u0647\u0627\u06cc $\\Delta \\hat{A}$ \u0648 $\\Delta \\hat{B}$ \u0631\u0627 \u0627\u06cc\u0646 \u0637\u0648\u0631 \u062a\u0639\u0631\u06cc\u0641 \u0645\u06cc \u06a9\u0646\u06cc\u0645:<\/p>\n

(2)<\/strong><\/p>\n

$\\large \\Delta\\hat{A}= \\hat{A}-\\langle \\hat{A} \\rangle \\; , \\; \\Delta\\hat{B}= \\hat{B}-\\langle \\hat{B} \\rangle $<\/p>\n

\u062a\u0648\u062c\u0647 \u06a9\u0646\u06cc\u062f \u06a9\u0647 \u0645\u0642\u062f\u0627\u0631 \u0686\u0634\u0645\u062f\u0627\u0634\u062a\u06cc \u06cc\u06a9 \u0627\u0633\u06a9\u0627\u0644\u0631 \u0627\u0633\u062a\u061b \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0645\u0631\u0628\u0639 \u0639\u0645\u0644\u06af\u0631\u0647\u0627\u06cc \u0628\u0627\u0644\u0627 \u0639\u0628\u0627\u0631\u062a \u067e\u0627\u06cc\u06cc\u0646 \u0627\u0633\u062a:<\/p>\n

(3)<\/strong><\/p>\n

$\\large (\\Delta\\hat{A})^2=\\hat{A}^2-2\\hat{A}\\langle \\hat{A} \\rangle+\\langle \\hat{A} \\rangle^2$<\/p>\n

$\\large (\\Delta\\hat{B})^2=\\hat{B}^2-2\\hat{B}\\langle \\hat{B} \\rangle+\\langle \\hat{B} \\rangle^2$<\/p>\n

\u062d\u0627\u0644 \u0645\u0642\u062f\u0627\u0631 \u0686\u0634\u0645\u062f\u0627\u0634\u062a\u06cc (\u0646\u0633\u0628\u062a \u0628\u0647 \u0628\u0631\u062f\u0627\u0631 \u062d\u0627\u0644\u062a \u0628\u0647\u0646\u062c\u0627\u0631 $| \\psi \\rangle $) \u0639\u0645\u0644\u06af\u0631\u0647\u0627\u06cc \u0628\u0627\u0644\u0627 \u0631\u0627 \u0628\u062f\u0633\u062a \u0645\u06cc \u0622\u0648\u0631\u0645:<\/p>\n

(4)<\/strong><\/p>\n

$\\langle (\\Delta\\hat{A})^2 \\rangle = \\langle \\hat{A}^2 \\rangle – 2 \\langle \\hat{A} \\langle \\hat{A} \\rangle \\rangle + \\langle \\langle \\hat{A} \\rangle^2 \\rangle$<\/p>\n

$\\langle (\\Delta\\hat{B})^2 \\rangle = \\langle \\hat{B}^2 \\rangle – 2 \\langle \\hat{B} \\langle \\hat{B} \\rangle \\rangle + \\langle \\langle \\hat{B} \\rangle^2 \\rangle$<\/p>\n

\u0628\u0627\u0632\u0647\u0645 \u06cc\u0627\u062f\u0622\u0648\u0631\u06cc \u0645\u06cc \u06a9\u0646\u0645 \u06a9\u0647 \u0645\u0642\u062f\u0627\u0631 \u0686\u0634\u0645\u062f\u0627\u0634\u062a\u06cc \u0627\u0633\u06a9\u0627\u0644\u0631\u06cc \u0628\u06cc\u0634 \u0646\u06cc\u0633\u062a \u0648 \u0645\u06cc \u062a\u0648\u0627\u0646\u062f \u0627\u0632 $\\langle\u00a0 \\rangle$ \u062e\u0627\u0631\u062c \u0634\u0648\u062f\u061b \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u062f\u0631 \u062c\u0645\u0644\u0647 \u062f\u0648\u0645 \u0633\u0645\u062a \u0631\u0627\u0633\u062a \u062a\u0633\u0627\u0648\u06cc \u0647\u0627\u060c \u0645\u0631\u0628\u0639 \u0645\u0642\u062f\u0627\u0631 \u0686\u0634\u0645\u062f\u0627\u0634\u062a\u06cc \u0638\u0627\u0647\u0631 \u0645\u06cc \u0634\u0648\u062f:<\/p>\n

(5)<\/strong><\/p>\n

$\\langle (\\Delta\\hat{A})^2 \\rangle = \\langle \\hat{A}^2 \\rangle – 2 \\langle \\hat{A} \\rangle^2 + \\langle \\hat{A} \\rangle^2 \\langle \\psi | \\psi \\rangle $<\/p>\n

$\\langle (\\Delta\\hat{B})^2 \\rangle = \\langle \\hat{B}^2 \\rangle – 2 \\langle \\hat{B} \\rangle^2 + \\langle \\hat{B} \\rangle^2 \\langle \\psi | \\psi \\rangle $<\/p>\n

\u0686\u0648\u0646 \u0628\u0631\u062f\u0627\u0631 \u062d\u0627\u0644\u062a \u0628\u0647\u0646\u062c\u0627\u0631 \u0627\u062e\u062a\u06cc\u0627\u0631 \u06a9\u0631\u062f\u06cc\u0645 $\\langle \\psi | \\psi \\rangle=1$ \u067e\u0633:<\/p>\n

(6)<\/strong><\/p>\n

$\\large \\langle (\\Delta\\hat{A})^2 \\rangle = \\langle \\hat{A}^2 \\rangle – \\langle \\hat{A} \\rangle^2$<\/p>\n

$\\large \\langle (\\Delta\\hat{B})^2 \\rangle = \\langle \\hat{B}^2 \\rangle – \\langle \\hat{B} \\rangle^2$<\/p>\n

\u0628\u0647 \u062c\u0630\u0631 \u0639\u0628\u0627\u0631\u062a \u0647\u0627\u06cc \u0628\u0627\u0644\u0627\u060c \u0628\u0647 \u062a\u0631\u062a\u06cc\u0628\u060c \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0639\u0645\u0644\u06af\u0631 \u0647\u0627\u06cc $A$ \u0648 $B$ \u06af\u0648\u06cc\u06cc\u0645 (\u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0647\u0627 \u0628\u0627 \u0639\u0645\u0644\u06af\u0631\u0647\u0627\u06cc \u0631\u0627\u0628\u0637\u0647 (2) \u062f\u0627\u0631\u0627\u06cc \u062a\u0641\u0627\u0648\u062a\u0646\u062f):<\/p>\n

(7)<\/strong><\/p>\n

$\\large \\Delta A =\\sqrt{ \\langle \\hat{A}^2 \\rangle – \\langle \\hat{A} \\rangle^2 }$<\/p>\n

$\\large \\Delta B =\\sqrt{ \\langle \\hat{B}^2 \\rangle – \\langle \\hat{B} \\rangle^2 }$<\/p>\n

2-9 \u0631\u0627\u0628\u0637\u0647 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a<\/h3>\n

\u0628\u0631\u0627\u06cc \u0645\u062d\u0627\u0633\u0628\u0647 \u0631\u0627\u0628\u0637\u0647 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a\u060c \u0646\u062e\u0633\u062a\u06cc\u0646 \u06af\u0627\u0645 \u0628\u0631\u0631\u0633\u06cc \u062a\u0627\u062b\u06cc\u0631 \u0639\u0645\u0644\u06af\u0631\u0647\u0627\u06cc \u0631\u0627\u0628\u0637\u0647 (2) \u0628\u0631 $| \\psi \\rangle $ \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f:<\/p>\n

(8)<\/strong><\/p>\n

$\\large | \\chi \\rangle = \\Delta \\hat{A} | \\psi \\rangle=(\\hat{A}-\\langle \\hat{A} \\rangle)\\; | \\psi \\rangle$<\/p>\n

$\\large | \\phi \\rangle = \\Delta \\hat{B} | \\psi \\rangle=(\\hat{B}-\\langle \\hat{B} \\rangle)\\; | \\psi \\rangle$<\/p>\n

\u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0627\u06cc\u0646\u06a9\u0647 $\\hat{A}$ \u0648 $\\hat{B}$ \u0647\u0631\u0645\u06cc\u062a\u06cc \u0627\u0646\u062f \u0648 \u0645\u0642\u062f\u0627\u0631\u0686\u0634\u0645\u062f\u0627\u0634\u062a\u06cc \u0639\u0645\u0644\u06af\u0631 \u0647\u0631\u0645\u06cc\u062a\u06cc\u060c \u062d\u0642\u06cc\u0642\u06cc \u0627\u0633\u062a\u061b $\\Delta \\hat{A}$ \u0648 $\\Delta \\hat{B}$ \u0646\u06cc\u0632 \u0647\u0631\u0645\u06cc\u062a\u06cc \u062e\u0648\u0627\u0647\u0646\u062f \u0628\u0648\u062f:<\/p>\n

(9)<\/strong><\/p>\n

$\\large (\\Delta \\hat{A})^\\dagger=\\hat{A}^\\dagger – \\langle \\hat{A} \\rangle^* = \\hat{A} – \\langle \\hat{A} \\rangle=\\Delta \\hat{A}$<\/p>\n

$\\large (\\Delta \\hat{B})^\\dagger=\\hat{B}^\\dagger – \\langle \\hat{B} \\rangle^* = \\hat{B} – \\langle \\hat{B} \\rangle=\\Delta \\hat{B}$<\/p>\n

\u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0633\u0647 \u0631\u0627\u0628\u0637\u0647 \u0632\u06cc\u0631 \u0628\u0631\u0642\u0631\u0627\u0631 \u0627\u0633\u062a:<\/p>\n

(10)<\/strong><\/p>\n

$\\large \\langle \\chi | \\chi \\rangle=\\langle \\psi | (\\Delta \\hat{A})^2 | \\psi \\rangle$<\/p>\n

$\\large \\langle \\phi | \\phi \\rangle=\\langle \\psi | (\\Delta \\hat{B})^2 | \\psi \\rangle$<\/p>\n

$\\large \\langle \\chi | \\phi \\rangle = \\langle \\psi | (\\Delta \\hat{A})(\\Delta \\hat{B}) | \\psi \\rangle$<\/p>\n

\u0627\u0632 \u0641\u0635\u0644 1 \u0631\u0627\u0628\u0637\u0647 6<\/a> \u060c \u0646\u0627\u0645\u0633\u0627\u0648\u06cc \u0634\u0648\u0627\u0631\u062a\u0632 \u0631\u0627 \u0628\u062e\u0627\u0637\u0631 \u062f\u0627\u0631\u06cc\u0645:<\/p>\n

(11)<\/strong><\/p>\n

$\\large \\langle \\chi | \\chi \\rangle \\langle \\phi | \\phi \\rangle \\geq |\\langle \\chi | \\phi\\rangle |^2$<\/p>\n

\u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0631\u0627\u0628\u0637\u0647 (10) \u0648 (11)\u060c \u0646\u0627\u0645\u0633\u0627\u0648\u06cc \u0634\u0648\u0627\u0631\u062a\u0632 \u0631\u0627 \u0628\u0631\u0627\u06cc \u0645\u0633\u0626\u0644\u0647 \u062e\u0648\u062f\u0645\u0627\u0646 \u0628\u0627\u0632\u0646\u0648\u06cc\u0633\u06cc \u0645\u06cc \u06a9\u0646\u06cc\u0645:<\/p>\n

(12)<\/strong><\/p>\n

$\\large \\langle (\\Delta \\hat{A})^2 \\rangle \\langle (\\Delta \\hat{B})^2 \\rangle \\geq |\\langle (\\Delta \\hat{A})(\\Delta \\hat{B}) \\rangle|^2$<\/p>\n

\u06cc\u06a9 \u0631\u0627\u0647\u0628\u0631\u062f \u06a9\u0627\u0631\u0622\u0645\u062f\u060c \u0646\u0648\u0634\u062a\u0646 $\\Delta \\hat{A} \\Delta \\hat{B}$ \u0628\u0647 \u0634\u06a9\u0644 \u0632\u06cc\u0631 \u0627\u0633\u062a \u06a9\u0647 \u0628\u0631\u0642\u0631\u0627\u0631\u06cc \u0622\u0646 \u0628\u0647 \u0633\u0627\u062f\u06af\u06cc \u0642\u0627\u0628\u0644 \u062a\u062d\u0642\u06cc\u0642 \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f:<\/p>\n

(13)<\/strong><\/p>\n

$\\large \\Delta \\hat{A}\\Delta \\hat{B}=\\displaystyle{\\frac{1}{2}[\\Delta\\hat{A},\\Delta\\hat{B}]+\\frac{1}{2}\\{\\Delta\\hat{A},\\Delta\\hat{B}\\}}$<\/p>\n

\u0647\u0645\u0686\u0646\u06cc\u0646 \u062f\u06cc\u06af\u0631 \u0631\u0627\u0628\u0637\u0647 \u0627\u06cc \u06a9\u0647 \u0628\u0647 \u0633\u0627\u062f\u06af\u06cc \u062f\u0633\u062a\u06cc\u0627\u0628\u06cc \u0627\u0633\u062a: $[\\Delta\\hat{A},\\Delta\\hat{B}]=[\\hat{A},\\hat{B}]$\u060c \u0645\u06cc \u062f\u0627\u0646\u06cc\u0645 \u06a9\u0647 $[\\hat{A},\\hat{B}]$ \u067e\u0627\u062f\u0647\u0631\u0645\u06cc\u062a\u06cc \u0627\u0633\u062a \u0648 $\\{\\Delta\\hat{A},\\Delta\\hat{B}\\}$ \u0647\u0631\u0645\u06cc\u062a\u06cc. \u0647\u0645\u0686\u0646\u06cc\u0646 \u0645\u0642\u062f\u0627\u0631 \u0686\u0634\u0645\u062f\u0627\u0634\u062a\u06cc \u0639\u0645\u0644\u06af\u0631 \u0647\u0631\u0645\u06cc\u062a\u06cc\u060c \u062d\u0642\u06cc\u0642\u06cc \u0627\u0633\u062a \u0648 \u0645\u0642\u062f\u0627\u0631 \u0686\u0634\u0645\u062f\u0627\u0634\u062a\u06cc \u0639\u0645\u0644\u06af\u0631 \u067e\u0627\u062f\u0647\u0631\u0645\u06cc\u062a\u06cc\u060c \u0645\u0648\u0647\u0648\u0645\u06cc \u0627\u0633\u062a. \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u062d\u0633\u0627\u0628 $|\\langle (\\Delta \\hat{A})(\\Delta \\hat{B}) \\rangle|^2$ \u0633\u0631\u0631\u0627\u0633\u062a \u0627\u0633\u062a:<\/p>\n

(14)<\/strong><\/p>\n

$|\\langle (\\Delta \\hat{A})(\\Delta \\hat{B}) \\rangle|^2=\\displaystyle{\\frac{1}{4}|\\langle[ \\hat{A}, \\hat{B}]\\rangle|^2+\\frac{1}{4}|\\langle\\{\\Delta\\hat{A},\\Delta\\hat{B}\\}\\rangle|^2}$<\/p>\n

\u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0627\u06cc\u0646\u06a9\u0647 \u062c\u0645\u0644\u0647 \u062f\u0648\u0645 \u0633\u0645\u062a \u0631\u0627\u0633\u062a \u062a\u0633\u0627\u0648\u06cc \u0647\u0645\u0648\u0627\u0631\u0647 \u0645\u062b\u0628\u062a \u0627\u0633\u062a\u061b \u062c\u0645\u0644\u0647 \u0627\u0648\u0644 \u0644\u0632\u0648\u0645\u0627 \u0627\u0632 \u0633\u0645\u062a \u0686\u067e \u062a\u0633\u0627\u0648\u06cc \u06a9\u0648\u0686\u06a9\u062a\u0631 \u0627\u0633\u062a. \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0645\u06cc \u062a\u0648\u0627\u0646 \u0646\u0627\u0645\u0633\u0627\u0648\u06cc (12) \u0631\u0627 \u0628\u0647 \u0635\u0648\u0631\u062a \u0632\u06cc\u0631 \u0628\u0627\u0632\u0646\u0648\u06cc\u0633\u06cc \u06a9\u0631\u062f:<\/p>\n

(15)<\/strong><\/p>\n

$\\large \\langle (\\Delta \\hat{A})^2 \\rangle \\langle (\\Delta \\hat{B})^2 \\rangle \\geq \\displaystyle{\\frac{1}{4}|\\langle[ \\hat{A}, \\hat{B}]\\rangle|^2}$<\/p>\n

\u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0646\u0627\u0645\u0633\u0627\u0648\u06cc \u0627\u0631\u0632\u0634\u0645\u0646\u062f \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u062f\u0648 \u0639\u0645\u0644\u06af\u0631\u060c \u0628\u0627 \u062c\u0630\u0631 \u0627\u0632 \u0646\u0627\u0645\u0633\u0627\u0648\u06cc \u0631\u0627\u0628\u0637\u0647 \u0628\u0627\u0644\u0627 \u0628\u062f\u0633\u062a \u0645\u06cc \u0622\u06cc\u062f:<\/p>\n

(16)<\/strong><\/p>\n

$ \\large \\Delta A \\Delta B \\geq \\displaystyle{\\frac{1}{2}|\\langle[ \\hat{A}, \\hat{B}]\\rangle|}$<\/p>\n

\u0631\u0627\u0628\u0637\u0647 \u0628\u0627\u0644\u0627 \u0646\u0642\u0634 \u0627\u0633\u0627\u0633\u06cc \u062f\u0631 \u0635\u0648\u0631\u062a \u0628\u0646\u062f\u06cc \u0645\u06a9\u0627\u0646\u06cc\u06a9 \u06a9\u0648\u0627\u0646\u062a\u0648\u0645\u06cc \u0627\u06cc\u0641\u0627 \u0645\u06cc \u06a9\u0646\u062f. \u0628\u0647\u0631\u0647 \u0628\u0631\u062f\u0646 \u0627\u0632 \u0627\u06cc\u0646 \u0631\u0627\u0628\u0637\u0647 \u0628\u0631\u0627\u06cc \u0639\u0645\u0644\u06af\u0631\u0647\u0627\u06cc \u0645\u06a9\u0627\u0646 \u0648 \u062a\u06a9\u0627\u0646\u0647\u060c \u0645\u0627 \u0631\u0627 \u0628\u0647 \u0631\u0648\u0627\u0628\u0637 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0647\u0627\u06cc\u0632\u0646\u0628\u0631\u06af \u0633\u0648\u0642 \u0645\u06cc \u062f\u0647\u062f \u06a9\u0647 \u0627\u0632 \u0631\u0648\u0627\u0628\u0637 \u0627\u0633\u0627\u0633\u06cc \u0645\u06a9\u0627\u0646\u06cc\u06a9 \u06a9\u0648\u0627\u0646\u062a\u0648\u0645\u06cc \u0627\u0633\u062a.<\/p>\n

3-9 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0647\u0627\u06cc\u0632\u0646\u0628\u0631\u06af<\/h3>\n

\u0631\u0627\u0628\u0637\u0647 (16) \u0631\u0627 \u0628\u0631\u0627\u06cc \u062f\u0648 \u0639\u0645\u0644\u06af\u0631 $\\hat{X}$ \u0648 $\\hat{P}_x$ \u0645\u062d\u0627\u0633\u0628\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u0645\u061b \u0627\u0632 \u0641\u0635\u0644 \u0642\u0628\u0644 \u0645\u06cc \u062f\u0627\u0646\u06cc\u0645 \u06a9\u0647:<\/p>\n

(17)<\/strong><\/p>\n

$\\large [\\hat{X},\\hat{P}_x]=i\\hbar\\hat{I}$<\/p>\n

\u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0631\u0627\u0628\u0637\u0647 (16) \u0628\u0631\u0627\u06cc \u062f\u0648 \u0639\u0645\u0644\u06af\u0631 \u0645\u06a9\u0627\u0646 \u0648 \u062a\u06a9\u0627\u0646\u0647\u060c \u0634\u06a9\u0644 \u0632\u06cc\u0631 \u0631\u0627 \u0628\u0647 \u062e\u0648\u062f \u0645\u06cc \u06af\u06cc\u0631\u062f$|i\\hbar|=\\hbar$:<\/p>\n

(18)<\/strong><\/p>\n

$\\large \\Delta x \\Delta p_x \\geq \\displaystyle{\\frac{\\hbar}{2}}$<\/p>\n

\u0628\u0631\u0627\u06cc \u0639\u062f\u0645 \u062a\u062f\u0627\u062e\u0644 \u0628\u0627 \u0639\u0645\u0644\u06af\u0631 \u062a\u0631\u062c\u06cc\u062d \u062f\u0627\u062f\u0645 \u0627\u0632 \u0627\u0633\u0627\u0645\u06cc \u06a9\u0648\u0686\u06a9 \u0628\u0631\u0627\u06cc \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0645\u06a9\u0627\u0646 \u0648 \u062a\u06a9\u0627\u0646\u0647 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0646\u0645. \u0646\u062a\u06cc\u062c\u0647 \u0628\u0647 \u0637\u0648\u0631 \u0645\u0634\u0627\u0628\u0647 \u0628\u0631\u0627\u06cc \u0645\u0648\u0644\u0641\u0647 \u0647\u0627\u06cc \u062f\u06cc\u06af\u0631 \u0646\u06cc\u0632 \u0642\u0627\u0628\u0644 \u0645\u062d\u0627\u0633\u0628\u0647 \u0627\u0633\u062a:<\/p>\n

(19)<\/strong><\/p>\n

$\\large \\Delta y \\Delta p_y \\geq \\displaystyle{\\frac{\\hbar}{2}} \\; , \\; \\Delta z \\Delta p_z \\geq \\displaystyle{\\frac{\\hbar}{2}}$<\/p>\n

4-9 \u062c\u0645\u0639 \u0628\u0646\u062f\u06cc \u0641\u0635\u0644<\/h3>\n

\u0631\u0627\u0628\u0637\u0647 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0631\u0627 \u0628\u062f\u0633\u062a \u0622\u0648\u0631\u062f\u06cc\u0645 \u0648 \u0627\u0635\u0644 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0647\u0627\u06cc\u0632\u0646\u0628\u0631\u06af \u0646\u06cc\u0632 \u0627\u0632 \u0622\u0646 \u0627\u0633\u062a\u062e\u0631\u0627\u062c \u0634\u062f.<\/p>\n

 <\/p>\n\n\n\n
\u062a\u0645\u0631\u06cc\u0646\u0627\u062a \u0641\u0635\u0644<\/span><\/strong><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

*1- \u062f\u0631 \u0645\u062a\u0646 \u0627\u06cc\u0646 \u0641\u0635\u0644\u060c \u0627\u062b\u0628\u0627\u062a \u0647\u0627\u06cc\u06cc \u0631\u0627 \u0627\u0646\u062c\u0627\u0645 \u0646\u062f\u0627\u062f\u0647 \u0627\u0645 \u06a9\u0647 \u062e\u0648\u0628 \u0627\u0633\u062a \u0628\u0631\u0627\u06cc \u062f\u0631\u06a9 \u0628\u0647\u062a\u0631 \u0645\u0637\u0644\u0628 \u06a9\u0627\u0645\u0644 \u0634\u0648\u062f.<\/p>\n","protected":false},"excerpt":{"rendered":"

\u0627\u0632 \u06a9\u0627\u0631\u0628\u0631\u062f\u0647\u0627\u06cc \u062c\u0628\u0631 \u062c\u0627\u0628\u062c\u0627\u06af\u0631\u060c \u0645\u062d\u0627\u0633\u0628\u0647 \u0631\u0627\u0628\u0637\u0647 \u06a9\u0644\u06cc \u0628\u0631\u0627\u06cc \u062d\u0627\u0635\u0644 \u0636\u0631\u0628 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0647\u0627\u06cc \u062f\u0648 \u0639\u0645\u0644\u06af\u0631 \u062f\u0644\u062e\u0648\u0627\u0647 \u0627\u0633\u062a. \u0631\u0627\u0628\u0637\u0647 \u0627\u06cc \u06a9\u0647 \u0631\u0648\u0627\u0628\u0637 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a \u0647\u0627\u06cc\u0632\u0646\u0628\u0631\u06af \u0631\u0627 \u0646\u062a\u06cc\u062c\u0647 \u0645\u06cc \u062f\u0647\u062f. 1-9 \u062a\u0639\u0631\u06cc\u0641 \u0645\u0642\u062f\u0627\u0631 \u0686\u0634\u0645\u062f\u0627\u0634\u062a\u06cc \u0639\u0645\u0644\u06af\u0631\u0647\u0627\u06cc \u0647\u0631\u0645\u06cc\u062a\u06cc $\\hat{A}$ \u0648 $\\hat{B}$\u060c \u0646\u0633\u0628\u062a \u0628\u0647 \u0628\u0631\u062f\u0627\u0631 \u062d\u0627\u0644\u062a \u0628\u0647\u0646\u062c\u0627\u0631\u00a0$| \\psi \\rangle $ \u0628\u0647 \u0635\u0648\u0631\u062a \u0632\u06cc\u0631 \u0627\u0633\u062a: (1) $\\large \\langle \\hat{A} \\rangle=\\langle \\psi […]<\/p>\n","protected":false},"author":1,"featured_media":320,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"https:\/\/cdn.imgurl.ir\/uploads\/l7504_csm_1_1a64fa1c5a.jpg","fifu_image_alt":"\u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a","footnotes":""},"categories":[5,26],"tags":[28,27],"class_list":["post-319","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physics","category-26","tag-28","tag-27"],"yoast_head":"\n9 \u0639\u062f\u0645 \u0642\u0637\u0639\u06cc\u062a - 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