Revision as of 01:10, 24 February 2006

Overview of Chapter 3

 2/24/06

key ideas on measurement

The state of a system is system is described by a wavefunction. $ψ(r, t)$ . The wavefunction evolves deterministically according to the Schrodinger equation. However, we give a probabilistic interpretation to the wave function that allows us to predict the measurement of a given physical quantity.
On the other hand, if we perform an experiment, the system will be in some state. How do we obtain as much information about this state.
Finally, we may wish to perform an experiment on a system in a given state; i.e., one that is prepared experimentally to have well-defined properties.

reminder from chapter 2

$\langle \vec{r} \rangle _ t = \int \vec{r} \ |\psi(\vec{r},t)|^2 d^3 r$

$\langle \vec{p} \rangle _ t = \int \vec{p} \ |\phi(\vec{p},t)|^2 d^3 p$

But we can also compute the expectation of $p$ in position space. This is essential if we want to be able to treat variables such as angular momentum, which involve both position and momentum: $\vec{L} = \vec{r} \times \vec{p}$

Here is a fundamental result which you should prove:

(3.3) $\langle p _x \rangle _ t = -i \hbar \int \psi^*(\vec{r},t) \partial _x \psi(\vec{r},t) d^3 r$

$\partial _x$ means partial with respect to $x$ . And henceforth, equation numbers such as 3.3 above will refer to the equation number in the text.

The general vector form of 3.3 is the following:

(3.4) $\langle \vec{p} \rangle _ t = -i \hbar \int \psi^*(\vec{r},t) \nabla \psi(\vec{r},t) d^3 r = \int \psi^*(\vec{r},t) \left( \hbar/i \nabla \right) \psi(\vec{r},t) d^3 r$

The reason I put brackets around the $\hbar /i \nabla$ is that we can consider this operator as being the position space representation of $\vec{p}$ .

(3.9)

Physical Quantities and Observables

vectors and operators

We're going to build up to the definition of an operator, but let's start with the idea of physical vectors. These are objects that have a direction and length (or magnitude). The force on an object for example. In a particular coordinate system we can represent this vector as a 3-tuple. E.g., $\vec{A} = (A_x, A_y, A_z)$ . So the 3-tuple is the representation of the vector in a particular coordinate system. In other coordinates there will be other tuples, but the vector itself has an existance independent of the coordinates used.

So the way to think of a vector equation such as $A \vec{x} = \vec{y}$ is that the object $A$ is mapping one vector into another. If the vectors have the same length, then $A$ can be represented by a square matrix. In the same way, we can think of functions as vectors too. Just as we add vectors component-wise:

$[\vec{x} + \vec{y}] _ i = x_i + y_i$

we add functions point-wise. If f and g are functions of one variable, then

$[f + g](x) = f (x) + g (x)$

The main difference between these two equations is that the functions, in effect, have an infinite number of components, one for each value of the real variable x!

Now we can consider operators acting not on finite

@@ Line 35: / Line 35: @@
 =Physical Quantities and Observables=
+===vectors and operators===
 We're going to build up to the definition of an operator, but
-let's start with the idea of physical vectors.  These are objects that have a direction and length (or magnitude).  The force on an object for example.  In a particular coordinate system we can represent this vector as a 3-tuple.  E.g., <math>\vec{A} = (A_x, A_y, A_z)</math>.
+let's start with the idea of physical vectors.  These are objects that have a direction and length (or magnitude).  The force on an object for example.  In a particular coordinate system we can represent this vector as a 3-tuple.  E.g., <math>\vec{A} = (A_x, A_y, A_z)</math>.  So the 3-tuple is the '''representation''' of the vector in a particular coordinate system.  In other coordinates there will be other tuples, but the vector itself has an existance independent of the coordinates used.
+So the way to think of a vector equation such as <math>A \vec{x} = \vec{y}</math> is that the object <math>A</math> is '''mapping''' one vector into another.  If the vectors have the same length, then  <math>A</math>  can be '''represented''' by a square matrix.  In the same way, we can think of functions as vectors too.  Just as we add vectors component-wise:
+<math> [\vec{x} + \vec{y}] _ i = x_i + y_i </math>
+we add functions point-wise.  If f and g are functions of one variable, then
+<math> [f + g](x) = f(x) + g(x) </math>
+The main difference between these two equations is that '''the functions''', in effect, '''have an infinite number of components''', one for each value of the real variable x!
+Now we can consider operators acting not on finite

Modern 2:Overview of Chapter 3

Revision as of 01:10, 24 February 2006

Contents

Overview of Chapter 3

key ideas on measurement

reminder from chapter 2

Physical Quantities and Observables

vectors and operators

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