Update 5_operators_QM.mb

src/5_operators_QM.mb

@@ -47,34 +47,23 @@ $$\hat{A}[c_1|\psi_1\rangle+c_2|\psi_2\rangle]=c_1\hat{A}|\psi_1\rangle+c_2\hat{
 
 Linearity of operators has an important consequence. Recall that in the previous lecture we discussed that any state vector $|\psi\rangle can be expressed as a linear combination of a complete set of basis states $\{|\phi_i\rangle,i=1,2,3,...,n\}$ associated to this Hilbert space:
 $$|\psi\rangle=\sum_{i=1}^nc_i|\phi_i\rangle \, , \quad  c_i = \braket{\phi_i}{\psi} \, ,$$
-where the values of the coefficients $c_i$ can be fixed thanks to
-the orthogonality properties of the basis, $\braket{\phi_i}{\phi_j}=\delta_{ij} $.
+where the values of the coefficients $c_i$ can be fixed thanks to the orthogonality properties of the basis, $\braket{\phi_i}{\phi_j}=\delta_{ij} $.
   
-Then one can see that for linear operators
-  one has
-$$
-  \hat{A}|\psi\rangle=  \hat{A}\sum_{i=1}^nc_i|\phi_i\rangle
-  =  \sum_{i=1}^nc_i ( \hat{A}|\phi_i\rangle ) \, .
+Then one can see that for linear operators one has
+$$ \hat{A}|\psi\rangle=  \hat{A}\sum_{i=1}^nc_i|\phi_i\rangle =  \sum_{i=1}^nc_i ( \hat{A}|\phi_i\rangle ) \, .
 $$
 This results tells us that if we know the effects of the operator
 $\hat{A}$ for each of the elements of the basis $|\phi_i\rangle$,
 we can easily determine its effects for a *general state vector}
 $|\psi\rangle$ belonging to the same Hilbert space.
 
-Another important properties of operators can be stated
-  as follows.
-%
-  If two operators $\hat{A}$ and $\hat{B}$ are such that
-    $$
-        \hat{A}|\psi\rangle=\hat{B}|\psi\rangle
-    $$
-    for all state vectors $|\psi\rangle$ belonging to the Hilbert
-    space of the system, then two operators must be identical''
-    $$
-        \hat{A}=\hat{B} \, .
-    $$
-    Note that this is true only if the action of two operators
-    is identical for all elements of the Hilbert space.
+Another important properties of operators can be stated as follows. If two operators $\hat{A}$ and $\hat{B}$ are such that
+$$\hat{A}|\psi\rangle=\hat{B}|\psi\rangle $$
+for all state vectors $|\psi\rangle$ belonging to the Hilbert
+space of the system, then two operators must be identical:
+$$ \hat{A}=\hat{B} \, .$$
+Note that this is true only if the action of two operators
+is identical for all elements of the Hilbert space.
     
   As in general vector spaces, in Hilbert spaces
     we also have the identity (or unit) and zero (or null)