{"id":328,"date":"2026-08-12T08:16:33","date_gmt":"2026-08-12T04:46:33","guid":{"rendered":"https:\/\/sciencio.ir\/?p=328"},"modified":"2026-08-12T08:16:33","modified_gmt":"2026-08-12T04:46:33","slug":"12-%d9%88%db%8c%da%98%d9%87-%d9%85%d9%82%d8%af%d8%a7%d8%b1-%d9%88-%d9%88%db%8c%da%98%d9%87-%d8%a8%d8%b1%d8%af%d8%a7%d8%b1-%d8%b9%d9%85%d9%84%da%af%d8%b1","status":"publish","type":"post","link":"https:\/\/sciencio.ir\/?p=328","title":{"rendered":"12 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0648 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 \u0639\u0645\u0644\u06af\u0631"},"content":{"rendered":"

\u062f\u0631 \u0641\u0635\u0648\u0644 \u0642\u0628\u0644 \u0639\u0645\u0644\u06af\u0631\u0647\u0627\u060c \u062d\u0627\u0644\u062a \u0647\u0627 \u0648 \u0648\u06cc\u0698\u06af\u06cc \u0647\u0627\u06cc \u0622\u0646 \u0647\u0627 \u0631\u0627 \u0628\u0631\u0631\u0633\u06cc \u0646\u0645\u0648\u062f\u06cc\u0645. \u0628\u062d\u062b \u0627\u06cc\u0646 \u0641\u0635\u0644 \u062f\u0631\u0628\u0627\u0631\u0647 \u0686\u06af\u0648\u0646\u06af\u06cc \u06cc\u0627\u0641\u062a\u0646 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0647\u0627 (Eigenvalues) \u0648 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631\u0647\u0627 (Eigenvectors) \u0628\u0631\u0627\u06cc \u06cc\u06a9 \u0639\u0645\u0644\u06af\u0631 \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f. \u0627\u0644\u0628\u062a\u0647 \u0648\u0627\u0698\u0647 \u0647\u0627\u06cc \u0645\u0642\u062f\u0627\u0631 \u0648\u06cc\u0698\u0647 \u0648 \u0628\u0631\u062f\u0627\u0631 \u0648\u06cc\u0698\u0647 \u0646\u06cc\u0632 \u0628\u0631\u0627\u06cc \u0647\u0631\u062f\u0648 \u0645\u062a\u062f\u0627\u0648\u0644 \u0627\u0633\u062a.<\/p>\n

1-12 \u062a\u0639\u0631\u06cc\u0641<\/h3>\n

\u0628\u0631\u062f\u0627\u0631 \u062d\u0627\u0644\u062a \u062e\u0627\u0635 $| \\psi \\rangle$ \u0631\u0627 \u062f\u0631 \u0646\u0638\u0631 \u0628\u06af\u06cc\u0631\u06cc\u062f\u061b \u0647\u0631\u06af\u0627\u0647 \u0639\u0645\u0644\u06af\u0631 $\\hat{A}$ \u0628\u0631 \u0622\u0646 \u0627\u0639\u0645\u0627\u0644 \u0634\u0648\u062f \u0648 \u0636\u0631\u06cc\u0628\u06cc \u0645\u062e\u062a\u0644\u0637 \u0645\u0627\u0646\u0646\u062f $a$ \u0627\u0632 $| \\psi \\rangle$ \u0631\u0627 \u0646\u062a\u06cc\u062c\u0647 \u062f\u0647\u062f\u061b \u0628\u0647 \u0627\u06cc\u0646 \u0628\u0631\u062f\u0627\u0631 \u062d\u0627\u0644\u062a\u061b \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 \u0639\u0645\u0644\u06af\u0631 $\\hat{A}$ \u0648 \u0628\u0647 \u0622\u0646 \u0636\u0631\u06cc\u0628 \u0645\u062e\u062a\u0644\u0637 $a$\u060c \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0639\u0645\u0644\u06af\u0631 $\\hat{A}$ \u0645\u06cc \u06af\u0648\u06cc\u0646\u062f:<\/p>\n

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

$\\large \\hat{A} | \\psi \\rangle = a | \\psi \\rangle$<\/p>\n

\u0645\u0639\u0627\u062f\u0644\u0647 \u0628\u0627\u0644\u0627 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0645\u0639\u0627\u062f\u0644\u0647 \u06cc\u0627 \u0645\u0633\u0626\u0644\u0647 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 $\\hat{A}$ \u0634\u0646\u0627\u062e\u062a\u0647 \u0645\u06cc \u0634\u0648\u062f.<\/p>\n

2-12 \u062e\u0648\u0627\u0635 \u0645\u0633\u0627\u0626\u0644 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631<\/h3>\n

\u0627\u0632 \u0622\u0646\u062c\u0627\u06cc\u06cc \u06a9\u0647 \u0639\u0645\u0644\u06af\u0631 \u06cc\u06a9\u0647 $\\hat{I}$ \u0628\u0631 \u0647\u0631 \u0628\u0631\u062f\u0627\u0631\u062d\u0627\u0644\u062a\u06cc \u0627\u062b\u0631 \u06a9\u0646\u062f \u0622\u0646 \u0631\u0627 \u0646\u062a\u06cc\u062c\u0647 \u0645\u06cc \u062f\u0647\u062f\u061b \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0639\u0645\u0644\u06af\u0631 \u06cc\u06a9\u0647 $1$ \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f. \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u062a\u0645\u0627\u0645\u06cc \u0628\u0631\u062f\u0627\u0631\u0647\u0627\u060c \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 \u0639\u0645\u0644\u06af\u0631 \u06cc\u06a9\u0647 \u0627\u0646\u062f:<\/p>\n

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

$\\large \\hat{I} | \\psi \\rangle = | \\psi \\rangle$<\/p>\n

\u0631\u0627\u0628\u0637\u0647 (1) \u0631\u0627 \u062f\u0631\u0646\u0638\u0631 \u0628\u06af\u06cc\u0631\u06cc\u062f\u061b \u0647\u0631 \u062a\u0648\u0627\u0646 \u0637\u0628\u06cc\u0639\u06cc $n$ \u0627\u0632 \u0639\u0645\u0644\u06af\u0631 $\\hat{A}$\u060c \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 $a^n$ \u0631\u0627 \u0628\u0627 \u0647\u0645\u0627\u0646 \u0628\u0631\u062f\u0627\u0631 \u0648\u06cc\u0698\u0647 \u062e\u0648\u0627\u0647\u062f \u062f\u0627\u0634\u062a.<\/p>\n

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

$\\large \\hat{A}^n | \\psi \\rangle = a^n | \\psi \\rangle$<\/p>\n

\u0627\u062b\u0628\u0627\u062a<\/h5>\n

\u0628\u0647 \u0633\u0627\u062f\u06af\u06cc \u0645\u06cc \u062a\u0648\u0627\u0646 \u0631\u0627\u0628\u0637\u0647 \u0628\u0627\u0644\u0627 \u0631\u0627 \u0627\u062b\u0628\u0627\u062a \u0646\u0645\u0648\u062f\u061b \u0627\u06af\u0631 \u06cc\u06a9 \u0639\u0645\u0644\u06af\u0631 $\\hat{A}$ \u062f\u0631 \u062f\u0648 \u0637\u0631\u0641 \u062a\u0633\u0627\u0648\u06cc \u0631\u0627\u0628\u0637\u0647 (1) \u0636\u0631\u0628 \u06a9\u0646\u06cc\u0645:<\/p>\n

$\\large \\hat{A}^2 | \\psi \\rangle = a \\hat{A} | \\psi \\rangle$<\/p>\n

\u06a9\u0647 \u0637\u0631\u0641 \u0631\u0627\u0633\u062a \u062a\u0633\u0627\u0648\u06cc \u0627\u0632 \u0631\u0627\u0628\u0637\u0647 (1) \u0645\u0639\u0627\u062f\u0644 \u0627\u0633\u062a \u0628\u0627 $ (a) a | \\psi \\rangle $. \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u062e\u0648\u0627\u0647\u06cc\u0645 \u062f\u0627\u0634\u062a:<\/p>\n

$\\large \\hat{A}^2 | \\psi \\rangle = a^2 | \\psi \\rangle$<\/p>\n

\u062d\u0627\u0644 \u0645\u06cc \u062a\u0648\u0627\u0646 \u0628\u0647 \u0637\u0648\u0631 \u0645\u0634\u0627\u0628\u0647 \u0647\u0631\u062a\u0639\u062f\u0627\u062f \u062f\u0644\u062e\u0648\u0627\u0647 $\\hat{A}$ \u062f\u0631 \u0631\u0627\u0628\u0637\u0647 \u0628\u0627\u0644\u0627 \u0636\u0631\u0628 \u06a9\u0631\u062f \u0648 (3) \u0631\u0627 \u0646\u062a\u06cc\u062c\u0647 \u06af\u0631\u0641\u062a.<\/p>\n

\n

\u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0627\u06cc\u0646\u06a9\u0647 \u062f\u0631 \u0641\u0635\u0644 10 – \u0631\u0627\u0628\u0637\u0647 (1)<\/a> \u060c\u062a\u0627\u0628\u0639 \u0639\u0645\u0644\u06af\u0631\u0647\u0627\u06cc \u06a9\u0648\u0627\u0646\u062a\u0648\u0645\u06cc \u0631\u0627 \u0628\u0647 \u0635\u0648\u0631\u062a \u0633\u0631\u06cc \u062a\u0648\u0627\u0646\u06cc \u0645\u0639\u0631\u0641\u06cc \u06a9\u0631\u062f\u06cc\u0645\u061b \u0645\u06cc \u062a\u0648\u0627\u0646 \u0627\u0632 (3) \u0646\u062a\u06cc\u062c\u0647 \u06af\u0631\u0641\u062a:<\/p>\n

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

$\\large F(\\hat{A}) | \\psi \\rangle = F(a) | \\psi \\rangle$<\/p>\n

\u0628\u0631\u0627\u06cc \u0645\u062b\u0627\u0644 \u062a\u0627\u0628\u0639 \u0646\u0645\u0627\u06cc\u06cc \u0628\u0627 \u0648\u0631\u0648\u062f\u06cc \u0639\u0645\u0644\u06af\u0631 $\\hat{A}$\u060c \u0645\u0633\u0626\u0644\u0647 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0632\u06cc\u0631 \u0631\u0627 \u062f\u0627\u0631\u062f:<\/p>\n

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

$\\large e^{\\hat{A}} | \\psi \\rangle = e^{a} | \\psi \\rangle$<\/p>\n

\u0627\u0632 \u0631\u0648\u06cc $\\hat{A}^{-1}\\hat{A}=\\hat{I}$ \u0648 \u0631\u0648\u0627\u0628\u0637 (1) \u0648 (2) \u0645\u06cc \u062a\u0648\u0627\u0646 \u0628\u0647 \u0633\u0627\u062f\u06af\u06cc \u067e\u06cc \u0628\u0631\u062f \u06a9\u0647 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 $\\hat{A}^{-1}$ \u0645\u0639\u06a9\u0648\u0633 $a$ \u0627\u0633\u062a \u0648 \u0647\u0645\u0686\u0646\u0627\u0646 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631\u0647\u0627 \u06cc\u06a9\u0633\u0627\u0646 \u0645\u06cc \u0645\u0627\u0646\u0646\u062f \u06a9\u0647 \u0627\u062b\u0628\u0627\u062a \u0622\u0646 \u0631\u0627 \u062f\u0631 \u062a\u0645\u0631\u06cc\u0646 1 \u0628\u0647 \u0634\u0645\u0627 \u0645\u06cc \u0633\u067e\u0627\u0631\u0645:<\/p>\n

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

$\\large \\hat{A}^{-1} | \\psi \\rangle = \\displaystyle{\\frac{1}{a}} | \\psi \\rangle$<\/p>\n

3-12 \u0642\u0636\u0627\u06cc\u0627\u06cc \u0645\u0631\u0628\u0648\u0637 \u0628\u0647 \u0645\u0633\u0627\u0626\u0644 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u06cc<\/h3>\n

\u062f\u0631 \u0627\u06cc\u0646\u062c\u0627 \u0686\u0646\u062f \u0642\u0636\u06cc\u0647 \u067e\u0631\u06a9\u0627\u0631\u0628\u0631\u062f \u0648 \u0645\u0647\u0645 \u0645\u0631\u0628\u0648\u0637 \u0628\u0647 \u0645\u0633\u0627\u0626\u0644 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u06cc \u0631\u0627 \u0645\u0648\u0631\u062f \u0628\u0631\u0631\u0633\u06cc \u0642\u0631\u0627\u0631 \u0645\u06cc \u062f\u0647\u06cc\u0645. \u0627\u06cc\u0646 \u0642\u0636\u0627\u06cc\u0627 \u0627\u0647\u0645\u06cc\u062a \u0642\u0627\u0628\u0644 \u062a\u0648\u062c\u0647\u06cc \u062f\u0631 \u0635\u0648\u0631\u062a \u0628\u0646\u062f\u06cc \u0631\u06cc\u0627\u0636\u06cc \u0645\u06a9\u0627\u0646\u06cc\u06a9 \u06a9\u0648\u0627\u0646\u062a\u0648\u0645\u06cc \u062e\u0648\u0627\u0647\u0646\u062f \u062f\u0627\u0634\u062a.<\/p>\n

\u0642\u0636\u06cc\u0647 1<\/h5>\n

\u0628\u0631\u0627\u06cc \u0647\u0631 \u0639\u0645\u0644\u06af\u0631 \u0647\u0631\u0645\u06cc\u062a\u06cc\u060c \u0647\u0645\u0647 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u0647\u0627 \u062d\u0642\u06cc\u0642\u06cc \u0627\u0646\u062f. \u0647\u0645\u0686\u0646\u06cc\u0646 \u0647\u0645\u0647 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631\u0647\u0627\u06cc \u0645\u062a\u0646\u0627\u0638\u0631 \u0628\u0627 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u0647\u0627\u06cc \u0645\u062a\u0641\u0627\u0648\u062a\u060c \u0628\u0631\u06cc\u06a9\u062f\u06cc\u06af\u0631 \u0639\u0645\u0648\u062f\u0646\u062f. \u0639\u0645\u0644\u06af\u0631 $\\hat{A}$ \u0631\u0627 \u06cc\u06a9 \u0639\u0645\u0644\u06af\u0631 \u0647\u0631\u0645\u06cc\u062a\u06cc ($\\hat{A}=\\hat{A}^{\\dagger}$) \u0648 $n$ \u0631\u0627 \u0632\u06cc\u0631\u0648\u0646\u062f(\u0627\u0646\u062f\u06cc\u0633) \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 \u0648 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0645\u062a\u0646\u0627\u0638\u0631 \u062f\u0631 \u0646\u0638\u0631 \u0645\u06cc \u06af\u06cc\u0631\u06cc\u0645:<\/p>\n

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

$\\large \\hat{A} | \\phi_n \\rangle = a_n | \\phi_n \\rangle \\Rightarrow a_n \\in \\mathbb{R} \\;$ \u0648<\/span> $\\large \\; \\langle \\phi_m | \\phi_n \\rangle = \\delta_{mn}$<\/p>\n

\u0627\u062b\u0628\u0627\u062a<\/h5>\n

\u0628\u0631\u0627\u06cc \u0627\u062b\u0628\u0627\u062a\u060c \u0627\u0628\u062a\u062f\u0627 \u0628\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0645\u0633\u0626\u0644\u0647 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0639\u0645\u0644\u06af\u0631 $\\hat{A}$ \u0648 $\\hat{A}^{\\dagger}$ \u0631\u0627 \u0645\u06cc\u0627\u0646 \u0628\u0631\u0627 \u0648 \u06a9\u062a \u0648\u06cc\u0698\u0647 \u062d\u0627\u0644\u062a \u0628\u0627 \u0627\u0646\u062f\u06cc\u0633 \u0647\u0627\u06cc \u0645\u062a\u0641\u0627\u0648\u062a \u0633\u0627\u0646\u062f\u0648\u06cc\u0686 \u06a9\u0646\u06cc\u0645:<\/p>\n

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

$\\hat{A} |\\phi_n \\rangle = a_n | \\phi_n \\rangle \\displaystyle{\\overset{\\langle \\phi_m |}{\\Rightarrow}} \\langle \\phi_m |\\hat{A} |\\phi_n \\rangle = a_n \\langle \\phi_m |\\phi_n \\rangle$<\/p>\n

$\\langle \\phi_m |\\hat{A}^{\\dagger} = a_m^* \\langle \\phi_m | \\displaystyle{\\overset{| \\phi_n \\rangle}{\\Rightarrow}} \\langle \\phi_m |\\hat{A}^{\\dagger} |\\phi_n \\rangle = a_m^* \\langle \\phi_m |\\phi_n \\rangle $<\/p>\n

\u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u0647\u0631\u0645\u06cc\u062a\u06cc \u0628\u0648\u062f\u0646 $\\hat{A}$ \u0637\u0631\u0641 \u0686\u067e \u0646\u062a\u0627\u06cc\u062c \u0628\u0627\u0644\u0627 \u0628\u0627\u0647\u0645 \u06cc\u06a9\u0633\u0627\u0646 \u0627\u0646\u062f. \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u06a9\u0627\u0641\u06cc \u0627\u0633\u062a \u062f\u0648 \u0631\u0627\u0628\u0637\u0647 \u0631\u0627 \u0627\u0632 \u0647\u0645 \u06a9\u0645 \u06a9\u0646\u06cc\u0645:<\/p>\n

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

$\\large (a_n – a_m^*) \\langle \\phi_m |\\phi_n \\rangle = 0$<\/p>\n

\u0631\u0627\u0628\u0637\u0647 \u0628\u0627\u0644\u0627 \u0631\u0627 \u062f\u0631 \u062f\u0648 \u062d\u0627\u0644\u062a \u0628\u0631\u0631\u0633\u06cc \u0645\u06cc \u06a9\u0646\u06cc\u0645:<\/p>\n

\u0627\u06af\u0631 $n=m$ \u0628\u0627\u0634\u062f:\u00a0<\/strong>\u0686\u0648\u0646 $\\langle \\phi_n |\\phi_n \\rangle > 0$ \u0627\u0633\u062a\u061b \u0628\u0627\u06cc\u062f \u067e\u0631\u0627\u0646\u062a\u0632 \u0635\u0641\u0631 \u0628\u0627\u0634\u062f \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 $a_n = a_n^*$ \u06a9\u0647 \u062a\u0639\u0631\u06cc\u0641 \u0627\u0639\u062f\u0627\u062f \u062d\u0642\u06cc\u0642\u06cc \u0627\u0633\u062a\u061b \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0647\u0645\u0647 \u0648\u06cc\u0698\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631 \u062d\u0642\u06cc\u0642\u06cc \u0627\u0646\u062f.<\/p>\n

\u0627\u06af\u0631 $n \\neq m$ \u0628\u0627\u0634\u062f:<\/strong> \u0628\u0647 \u0637\u0648\u0631 \u06a9\u0644\u06cc \u062f\u0644\u06cc\u0644\u06cc \u0646\u062f\u0627\u0631\u062f \u06a9\u0647 \u067e\u0631\u0627\u0646\u062a\u0632 \u0635\u0641\u0631 \u0628\u0627\u0634\u062f ($a_n \\neq a_m^*$)\u061b \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0644\u0627\u0632\u0645 \u0627\u0633\u062a \u06a9\u0647 $ \\langle \\phi_m |\\phi_n \\rangle = 0$ \u0628\u0631\u0642\u0631\u0627\u0631 \u0628\u0627\u0634\u062f \u06a9\u0647 \u0646\u062a\u06cc\u062c\u0647 \u062f\u0648\u0645 \u0642\u0636\u06cc\u0647 \u0631\u0627 \u0628\u0647 \u0645\u0627 \u0645\u06cc \u062f\u0647\u062f.<\/p>\n

\n
\u0642\u0636\u06cc\u0647 2<\/h5>\n

\u0648\u06cc\u0698\u0647 \u062d\u0627\u0644\u062a \u0647\u0627\u06cc \u06cc\u06a9 \u0639\u0645\u0644\u06af\u0631 \u0647\u0631\u0645\u06cc\u062a\u06cc\u060c \u0645\u062c\u0645\u0648\u0639\u0647 \u0627\u06cc \u06a9\u0627\u0645\u0644 \u0627\u0632 \u062d\u0627\u0644\u062a \u0647\u0627\u06cc \u067e\u0627\u06cc\u0647 \u062f\u0648\u0628\u0647\u200c\u062f\u0648 \u0645\u062a\u0639\u0627\u0645\u062f \u0628\u0647\u0646\u062c\u0627\u0631 \u0645\u06cc \u0633\u0627\u0632\u0646\u062f. \u0639\u0645\u0644\u06af\u0631 \u062f\u0631 \u0627\u06cc\u0646 \u0648\u06cc\u0698\u0647 \u067e\u0627\u06cc\u0647\u060c \u0642\u0637\u0631\u06cc \u0627\u0633\u062a \u0648 \u0639\u0646\u0627\u0635\u0631 \u0642\u0637\u0631\u06cc \u0647\u0645\u0627\u0646 \u0648\u06cc\u0698\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631\u0646\u062f. \u0632\u0645\u0627\u0646\u06cc \u0627\u06cc\u0646 \u0645\u062c\u0645\u0648\u0639\u0647 \u0627\u0632 \u067e\u0627\u06cc\u0647 \u0647\u0627 \u0645\u0646\u062d\u0635\u0631 \u0628\u0647 \u0641\u0631\u062f \u0627\u0633\u062a \u06a9\u0647 \u0639\u0645\u0644\u06af\u0631 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u0647\u0627\u06cc \u0648\u0627\u06af\u0646 (\u0648\u06cc\u0698\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631 \u062a\u06a9\u0631\u0627\u0631\u06cc) \u0646\u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u062f. \u062f\u0631 \u0635\u0648\u0631\u062a\u06cc \u06a9\u0647 \u0648\u06cc\u0698\u0647 \u0645\u0642\u0627\u062f\u06cc\u0631 \u0648\u0627\u06af\u0646 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u062f\u061b \u0628\u06cc \u0646\u0647\u0627\u06cc\u062a \u0648\u06cc\u0698\u0647 \u067e\u0627\u06cc\u0647 \u062e\u0648\u0627\u0647\u06cc\u0645 \u062f\u0627\u0634\u062a \u0648 \u0645\u0646\u062d\u0635\u0631 \u0628\u0647 \u0641\u0631\u062f \u0646\u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f.<\/p>\n

\u0642\u0636\u06cc\u0647 3<\/h5>\n

\u0647\u0631\u06af\u0627\u0647 \u062f\u0648 \u0639\u0645\u0644\u06af\u0631 \u0647\u0631\u0645\u06cc\u062a\u06cc $\\hat{A}$ \u0648 $\\hat{B}$ \u062c\u0627\u0628\u062c\u0627 \u0634\u0648\u0646\u062f ($\\hat{A}\\hat{B}=\\hat{B}\\hat{A}$) \u0648 \u0647\u0645\u0686\u0646\u06cc\u0646 $\\hat{A}$ \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0648\u0627\u06af\u0646 \u0646\u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u062f\u061b \u0622\u0646\u06af\u0627\u0647 \u0647\u0631 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 $\\hat{A}$\u060c \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 $\\hat{B}$ \u0646\u06cc\u0632 \u0647\u0633\u062a. \u0639\u0644\u0627\u0648\u0647 \u0628\u0631 \u0627\u06cc\u0646 \u0645\u06cc \u062a\u0648\u0627\u0646 \u067e\u0627\u06cc\u0647 \u0645\u062a\u0639\u0627\u0645\u062f \u0628\u0647\u0646\u062c\u0627\u0631 \u0645\u0634\u062a\u0631\u06a9\u06cc \u0633\u0627\u062e\u062a \u06a9\u0647 \u0627\u0632 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631\u0647\u0627\u06cc \u0645\u0634\u062a\u0631\u06a9 \u0627\u06cc\u0646 \u062f\u0648 \u0639\u0645\u0644\u06af\u0631 \u0633\u0627\u062e\u062a\u0647 \u0634\u062f\u0647 \u0628\u0627\u0634\u062f.<\/p>\n

\u0627\u062b\u0628\u0627\u062a<\/h5>\n

\u0628\u0647 \u0645\u0633\u0626\u0644\u0647 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u06cc $\\hat{A}$ \u0628\u0627\u0632 \u0645\u06cc \u06af\u0631\u062f\u06cc\u0645:<\/p>\n

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

$\\large \\hat{A} |\\phi_n \\rangle = a_n | \\phi_n \\rangle$<\/p>\n

\u0627\u0632 \u0633\u0645\u062a \u0686\u067e $\\hat{B}$ \u0631\u0627 \u062f\u0631 \u0645\u0639\u0627\u062f\u0644\u0647 \u0636\u0631\u0628 \u0645\u06cc \u06a9\u0646\u06cc\u0645 \u0648 \u062f\u0631\u0646\u0638\u0631 \u0645\u06cc \u06af\u06cc\u0631\u06cc\u0645 \u06a9\u0647 \u062f\u0648 \u0639\u0645\u0644\u06af\u0631 \u0628\u0627\u0647\u0645 \u062c\u0627\u0628\u062c\u0627 \u0645\u06cc \u0634\u0648\u0646\u062f:<\/p>\n

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

$\\large \\hat{B} \\hat{A} |\\phi_n \\rangle = \\hat{B} a_n | \\phi_n \\rangle$<\/p>\n

$\\large \\hat{A} (\\hat{B} |\\phi_n \\rangle) =a_n (\\hat{B} | \\phi_n \\rangle)$<\/p>\n

\u0631\u0627\u0628\u0637\u0647 \u0628\u0627\u0644\u0627 \u0646\u062a\u06cc\u062c\u0647 \u0645\u06cc \u062f\u0647\u062f \u06a9\u0647 $\\hat{B} | \\phi_n \\rangle$ \u0646\u06cc\u0632 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631\u06cc \u0628\u0631\u0627\u06cc \u0639\u0645\u0644\u06af\u0631 $\\hat{A}$ \u0628\u0627 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 $a_n$ \u0627\u0633\u062a. \u0686\u0648\u0646 \u062f\u0631 \u0646\u0638\u0631 \u06af\u0631\u0641\u062a\u06cc\u0645 \u06a9\u0647 \u0648\u06cc\u0698\u0647 \u062d\u0627\u0644\u062a \u0648\u0627\u06af\u0646 \u0646\u06cc\u0633\u062a\u061b \u0628\u0647 \u0627\u0632\u0627\u06cc \u0647\u0631 \u0648\u06cc\u0698\u0647 \u062d\u0627\u0644\u062a \u062a\u0646\u0647\u0627 \u06cc\u06a9 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 \u0648\u062c\u0648\u062f \u062e\u0648\u0627\u0647\u062f \u062f\u0627\u0634\u062a\u061b \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 $\\hat{B} | \\phi_n \\rangle$ \u062a\u0646\u0647\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646\u062f \u0636\u0631\u06cc\u0628\u06cc \u0645\u0627\u0646\u0646\u062f $b_n$ \u0627\u0632 $ | \\phi_n \\rangle$ \u0628\u0627\u0634\u062f. \u0628\u062f\u06cc\u0646 \u062a\u0631\u062a\u06cc\u0628 \u062e\u0648\u0627\u0647\u06cc\u0645 \u062f\u0627\u0634\u062a:<\/p>\n

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

$\\large \\hat{B} |\\phi_n \\rangle = b_n | \\phi_n \\rangle$<\/p>\n

\u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0647\u0631 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 $\\hat{A}$ \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 $\\hat{B}$ \u0646\u06cc\u0632 \u0647\u0633\u062a \u0648 \u0628\u0627\u0644\u0639\u06a9\u0633. \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0647\u0631\u062f\u0648 \u0639\u0645\u0644\u06af\u0631 \u06cc\u06a9 \u067e\u0627\u06cc\u0647 \u0645\u0634\u062a\u0631\u06a9 \u0645\u0646\u062d\u0635\u0631\u0628\u0647 \u0641\u0631\u062f \u062f\u0627\u0631\u0646\u062f \u06a9\u0647 \u0627\u0632 \u0627\u06cc\u0646 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631\u0647\u0627 \u0633\u0627\u062e\u062a\u0647 \u0634\u062f\u0647 \u0648 \u062f\u0631 \u0622\u0646 \u0647\u0631\u062f\u0648 \u0639\u0645\u0644\u06af\u0631 \u0642\u0637\u0631\u06cc \u062e\u0648\u0627\u0647\u0646\u062f \u0628\u0648\u062f. \u0627\u06cc\u0646 \u0642\u0636\u06cc\u0647 \u0628\u0631\u0627\u06cc \u0647\u0631\u062a\u0639\u062f\u0627\u062f \u0639\u0645\u0644\u06af\u0631 \u0647\u0631\u0645\u06cc\u062a\u06cc \u06a9\u0647 \u062f\u0648\u0628\u0647\u200c\u062f\u0648 \u062c\u0627\u0628\u062c\u0627 \u0634\u0648\u0646\u062f \u0628\u0631\u0642\u0631\u0627\u0631 \u0627\u0633\u062a.<\/p>\n

\n
\u0642\u0636\u06cc\u0647 4<\/h5>\n

\u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u0647\u0627\u06cc \u06cc\u06a9 \u0639\u0645\u0644\u06af\u0631 \u067e\u0627\u062f\u0647\u0631\u0645\u06cc\u062a\u06cc\u060c \u0645\u0648\u0647\u0648\u0645\u06cc \u0645\u062d\u0636 \u06cc\u0627 \u0635\u0641\u0631\u0646\u062f. \u0627\u06cc\u0646 \u0645\u0648\u0631\u062f \u0627\u062b\u0628\u0627\u062a\u06cc \u0645\u0634\u0627\u0628\u0647 \u0642\u0636\u06cc\u0647 1 \u062f\u0627\u0631\u062f \u06a9\u0647 \u062f\u0631 \u0634\u0631\u0637 $n=m$\u060c \u0628\u0647 \u0646\u062a\u06cc\u062c\u0647 $a_n = – a_n^*$ \u0645\u06cc \u0631\u0633\u06cc\u0645 \u06a9\u0647 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u062a\u0645\u0631\u06cc\u0646 \u062f\u0648\u0645 \u0627\u06cc\u0646 \u0641\u0635\u0644 \u0622\u0648\u0631\u062f\u0645.<\/p>\n

\u0642\u0636\u06cc\u0647 5<\/h5>\n

\u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u0647\u0627\u06cc \u06cc\u06a9 \u0639\u0645\u0644\u06af\u0631 \u06cc\u06a9\u0627\u0646\u06cc\u060c \u0627\u0639\u062f\u0627\u062f \u0645\u062e\u062a\u0644\u0637\u06cc \u0627\u0646\u062f \u06a9\u0647 \u0627\u0646\u062f\u0627\u0632\u0647 \u0622\u0646 \u0647\u0627 \u0628\u0631\u0627\u0628\u0631 \u0628\u0627 \u06cc\u06a9 \u0627\u0633\u062a. \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631\u0647\u0627\u06cc \u0639\u0645\u0644\u06af\u0631 \u06cc\u06a9\u0627\u0646\u06cc \u06a9\u0647 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u0647\u0627\u06cc \u0648\u0627\u06af\u0646 \u0646\u062f\u0627\u0631\u062f\u061b \u0628\u0647 \u0635\u0648\u0631\u062a \u062f\u0648\u0628\u0647\u200c\u062f\u0648 \u0645\u062a\u0639\u0627\u0645\u062f\u0646\u062f.<\/p>\n

\u0627\u062b\u0628\u0627\u062a<\/h5>\n

\u0645\u0633\u0626\u0644\u0647 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u06cc \u0639\u0645\u0644\u06af\u0631 \u06cc\u06a9\u0627\u0646\u06cc $\\hat{U}$ \u0628\u0647 \u0635\u0648\u0631\u062a \u0632\u06cc\u0631 \u0627\u0633\u062a:<\/p>\n

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

$\\large \\hat{U} |\\phi_n \\rangle = a_n | \\phi_n \\rangle$<\/p>\n

\u0627\u0644\u062d\u0627\u0642\u06cc \u0647\u0631\u0645\u06cc\u062a\u06cc \u0631\u0627\u0628\u0637\u0647 \u0628\u0627\u0644\u0627 \u0628\u0627 \u0627\u0646\u062f\u06cc\u0633 \u0645\u062a\u0641\u0627\u0648\u062a \u0631\u0627 \u062f\u0631 \u0646\u0638\u0631 \u0645\u06cc \u06af\u06cc\u0631\u06cc\u0645:<\/p>\n

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

$\\large \\langle \\phi_m | \\hat{U}^{\\dagger} = a_m^* \\langle \\phi_m |$<\/p>\n

\u062f\u0648 \u0637\u0631\u0641 \u062a\u0633\u0627\u0648\u06cc (14) \u0631\u0627 \u062f\u0631 (13) \u0627\u0639\u0645\u0627\u0644 \u0645\u06cc \u06a9\u0646\u06cc\u0645 ($\\hat{U}^\\dagger \\hat{U} = \\hat{I}$):
\n(15)<\/strong><\/p>\n

$\\large \\langle \\phi_m | \\phi_n \\rangle = a_m^* a_n \\langle \\phi_m | \\phi_n \\rangle$<\/p>\n

\u06a9\u0647 \u0628\u0647 \u0635\u0648\u0631\u062a \u0631\u0627\u0628\u0637\u0647 \u0632\u06cc\u0631 \u0642\u0627\u0628\u0644 \u0628\u0627\u0632\u0646\u0648\u06cc\u0633\u06cc \u0627\u0633\u062a:<\/p>\n

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

$\\large (a_m^* a_n – 1) \\langle \\phi_m | \\phi_n \\rangle = 0$<\/p>\n

\u062f\u0631 \u062d\u0627\u0644\u062a $n=m$<\/strong>\u060c \u0645\u0634\u0627\u0628\u0647 \u0642\u0628\u0644 $\\langle \\phi_n |\\phi_n \\rangle > 0$ \u0627\u0633\u062a\u061b \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u0628\u0627\u06cc\u062f \u067e\u0631\u0627\u0646\u062a\u0632 \u0635\u0641\u0631 \u0628\u0627\u0634\u062f. \u0627\u0632 \u0627\u06cc\u0646 \u0631\u0648 \u0628\u0627\u06cc\u062f $a_n^* a_n = 1$ \u0628\u0631\u0642\u0631\u0627\u0631 \u0628\u0627\u0634\u062f \u06a9\u0647 \u0646\u062a\u06cc\u062c\u0647 \u0645\u06cc \u062f\u0647\u062f $|a_n|=1$.<\/p>\n

\u062f\u0631 \u062d\u0627\u0644\u062a<\/strong> $n \\neq m$<\/strong>\u060c \u0628\u0627\u06cc\u062f $\\langle \\phi_m |\\phi_n \\rangle = 0$ \u0628\u0631\u0642\u0631\u0627\u0631 \u0628\u0627\u0634\u062f \u06a9\u0647 \u0646\u0634\u0627\u0646 \u062f\u0647\u0646\u062f\u0647 \u0645\u062a\u0639\u0627\u0645\u062f \u0628\u0648\u062f\u0646 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631\u0647\u0627\u0633\u062a.<\/p>\n

\n

4-12 \u062c\u0645\u0639 \u0628\u0646\u062f\u06cc \u0641\u0635\u0644<\/h3>\n

\u0628\u0627 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0648 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 \u06cc\u06a9 \u0639\u0645\u0644\u06af\u0631\u060c \u0645\u0633\u0627\u0626\u0644 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631\u06cc\u060c \u062e\u0648\u0627\u0635 \u0622\u0646 \u0648 \u0686\u0646\u062f \u0642\u0636\u06cc\u0647 \u0645\u0647\u0645 \u062f\u0631 \u0627\u06cc\u0646 \u0645\u0633\u0627\u0626\u0644 \u0622\u0634\u0646\u0627 \u0634\u062f\u06cc\u0645.<\/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- \u0631\u0627\u0628\u0637\u0647 (6) \u0631\u0627 \u0627\u062b\u0628\u0627\u062a \u0646\u0645\u0627\u06cc\u06cc\u062f.<\/p>\n

*2- \u0642\u0636\u06cc\u0647 4 \u0631\u0627 \u0627\u062b\u0628\u0627\u062a \u06a9\u0646\u06cc\u062f.<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

\u062f\u0631 \u0641\u0635\u0648\u0644 \u0642\u0628\u0644 \u0639\u0645\u0644\u06af\u0631\u0647\u0627\u060c \u062d\u0627\u0644\u062a \u0647\u0627 \u0648 \u0648\u06cc\u0698\u06af\u06cc \u0647\u0627\u06cc \u0622\u0646 \u0647\u0627 \u0631\u0627 \u0628\u0631\u0631\u0633\u06cc \u0646\u0645\u0648\u062f\u06cc\u0645. \u0628\u062d\u062b \u0627\u06cc\u0646 \u0641\u0635\u0644 \u062f\u0631\u0628\u0627\u0631\u0647 \u0686\u06af\u0648\u0646\u06af\u06cc \u06cc\u0627\u0641\u062a\u0646 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0647\u0627 (Eigenvalues) \u0648 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631\u0647\u0627 (Eigenvectors) \u0628\u0631\u0627\u06cc \u06cc\u06a9 \u0639\u0645\u0644\u06af\u0631 \u062e\u0648\u0627\u0647\u062f \u0628\u0648\u062f. \u0627\u0644\u0628\u062a\u0647 \u0648\u0627\u0698\u0647 \u0647\u0627\u06cc \u0645\u0642\u062f\u0627\u0631 \u0648\u06cc\u0698\u0647 \u0648 \u0628\u0631\u062f\u0627\u0631 \u0648\u06cc\u0698\u0647 \u0646\u06cc\u0632 \u0628\u0631\u0627\u06cc \u0647\u0631\u062f\u0648 \u0645\u062a\u062f\u0627\u0648\u0644 \u0627\u0633\u062a. 1-12 \u062a\u0639\u0631\u06cc\u0641 \u0628\u0631\u062f\u0627\u0631 \u062d\u0627\u0644\u062a \u062e\u0627\u0635 $| \\psi \\rangle$ \u0631\u0627 […]<\/p>\n","protected":false},"author":1,"featured_media":329,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"https:\/\/cdn.imgurl.ir\/uploads\/c262472_24368908_0_prlq.jpg","fifu_image_alt":"\u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631","footnotes":""},"categories":[5,26],"tags":[28,27],"class_list":["post-328","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physics","category-26","tag-28","tag-27"],"yoast_head":"\n12 \u0648\u06cc\u0698\u0647 \u0645\u0642\u062f\u0627\u0631 \u0648 \u0648\u06cc\u0698\u0647 \u0628\u0631\u062f\u0627\u0631 \u0639\u0645\u0644\u06af\u0631 - 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