Quantum Mechanics - Revision history

KevinYager: /* Density Matrices */

2021-06-16T20:45:26Z

‎Density Matrices

2021-06-16T20:36:26Z

‎Density Matrices

2021-06-16T20:31:32Z

‎Measurement

2021-06-16T20:25:06Z

‎Measurement

2021-06-16T20:22:04Z

‎Measurement

2021-06-16T20:12:53Z

‎Measurement

2021-06-16T20:12:12Z

‎Measurement

2021-06-16T19:56:58Z

‎Measurement

2021-06-16T19:43:03Z

‎Density Matrices

2015-02-25T18:01:25Z

‎Superposition

@@ Line 322: / Line 322: @@
 \end{bmatrix}
 </math>
-The off-diagonal terms indicate the strength and nature of the interference between the states. During decoherence, these off-diagonal terms are driven towards zero.
+The off-diagonal terms indicate the strength and nature of the interference between the states. Thus, the diagonal elements (<math>n=m</math>) capture the probability of occupying a certain state as <math>\rho_{nm} = \overline{c_n c_n^*} = p_n \geq 0 </math>The time-dependence often appears in these off-diagonal (<math>n\neq m</math>) terms as <math>\rho_{nm}(t) = \overline{c_n(t)c_m^*(t)} = \overline{c_n c_m^*} e^{-i \omega_{nm} t} </math>.
 ==Decoherence==

@@ Line 296: / Line 296: @@
 \rho(t) \equiv | \psi(t) \rangle \langle \psi(t) |
 </math>
-For instance, a '''pure state''' is given by a density matrix that has only a single non-zero term, along the diagonal:
+The elements of the density matrix can be interpreted as a form of quantum probability distribution. For instance, a '''pure state''' is given by a density matrix that has only a single non-zero term, along the diagonal:
 :<math>
 | \psi \rangle \langle \psi | =

@@ Line 198: / Line 198: @@
 |-
 | &nbsp;<math> f_{\phi} \in V^* </math>&nbsp;
-| &nbsp;<math> \langle \phi | </math>&nbsp;
+| &nbsp;<math> \langle \phi | = a_1^* \langle \phi_1 | + a_2^* \langle \phi_2 | + a_3^* \langle \phi_3 | + \dots </math>&nbsp;
 | &nbsp;<math> \langle \phi | = \begin{bmatrix} a_1^* & a_2^* & \dots & a_n^* \end{bmatrix} </math>&nbsp;

@@ Line 202: / Line 202: @@
 |}
-The linear map <math>\langle \phi |</math> is a function (in the dual space <math>V^*</math>) that maps from vector space <math>V</math> to <math>\mathbb{C}</math>. In conventional notation this operation would be performed using a function <math>f_{\phi}</math> applied to <math>\psi</math>:
+The linear map <math>\langle \phi |</math> is a function (in the complex vector space <math>V^*</math>, dual to <math>V</math>) that maps from vector space <math>V</math> to <math>\mathbb{C}</math>. In conventional notation this operation would be performed using a function <math>f_{\phi}</math> applied to <math>\psi</math>:
 :<math>f_{\phi}(\psi) = a_1^* \psi_1 + a_2^* \psi_2 + \dots + a_n^* \psi_n</math>

@@ Line 180: / Line 180: @@
 ==Measurement==
-For state <math>\psi</math> and linear map <math>\angle \phi |</math>:
+For state <math>\psi</math> and linear map <math>\langle \phi |</math>:
 {| class="wikitable"
@@ Line 202: / Line 202: @@
 |}
-The transition of the wavefunction <math>\psi</math> into state <math>\phi</math> can be thought of as applying <math>f_{\phi}</math> to <math>\psi</math> (computing <math>f_{\phi}(\psi)</math>). That is:
+:<math>f_{\phi}(\psi) = a_1^* \psi_1 + a_2^* \psi_2 + \dots + a_n^* \psi_n</math>
 {| class="wikitable"
 | &nbsp;<math> \int \phi^* \psi \mathrm{d}x </math>&nbsp;

@@ Line 266: / Line 266: @@
 \phi_N \psi_1^* & \phi_N \psi_2^* & \cdots & \phi_N \psi_N^* \end{bmatrix}
 </math>
-Density matrices can be a useful way to visualize the interactions between states of a system. For instance, a '''pure state''' is given by a density matrix that has only a single non-zero term, along the diagonal:
+Density matrices can be a useful way to visualize the interactions between states of a system, and time-evolution. Thus we can define the density matrix as:
 :<math>
 | \psi \rangle \langle \psi | =

@@ Line 118: / Line 118: @@
 ! Integral Notation
 ! Dirac Notation
 |-
-| &nbsp;<math> \psi(x) = \sum_n c_n \psi_n </math>&nbsp;
+| &nbsp;<math> \psi = \sum_n c_n \psi_n </math>&nbsp;
-| &nbsp;<math> |\psi\rangle = c_1 | 1 \rangle + c_2 | 2 \rangle + c_3 | 3 \rangle + \cdots </math>&nbsp;
+| &nbsp;<math>
 |}