Modern 2:Overview of Chapter 3
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− | + | == Overview of Chapter 3 == | |
+ | 2/24/06 | ||
* The state of a system is system is described by a wavefunction. <math>\psi(r,t)</math>. 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. | * The state of a system is system is described by a wavefunction. <math>\psi(r,t)</math>. 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. | ||
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'''Reminder''' | '''Reminder''' | ||
− | <math> \langle r \rangle _ t = \int r |\psi(r,t)|^2 d^3 r </math> | + | <math> \langle \vec{r} \rangle _ t = \int \vec{r} \ |\psi(\vec{r},t)|^2 d^3 r</math> |
− | <math> \langle p \rangle _ t = \int p |\phi(p,t)|^2 d^3 p </math> | + | <math> \langle \vec{p} \rangle _ t = \int \vec{p} |\phi(\vec{p},t)|^2 d^3 p </math> |
− | But we can also compute the expectation of <math>p</math> 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: <math>L = r \times p</math> | + | But we can also compute the expectation of <math>p</math> 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: <math>\vec{L} = \vec{r} \times \vec{p}</math> |
Here is a fundamental result which you should prove: | Here is a fundamental result which you should prove: | ||
− | <math> \langle p _x \rangle _ t = \int \psi^*(r,t) \partial _x \psi(r,t) d^3 r </math> | + | <math> \langle \vec{p} _x \rangle _ t = -i \hbar \int \psi^*(\vec{r},t) \partial _x \psi(\vec{r},t) d^3 r </math> |
+ | |||
+ | <math> \partial _x </math> means partial with respect to <math>x</math>. |
Revision as of 00:18, 24 February 2006
Overview of Chapter 3
2/24/06
- 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
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:
Here is a fundamental result which you should prove:
means partial with respect to x.