Hi there,

While you're waiting for an answer, did you know we have 300,000 study resources that could answer your question in TSR's Learn together section?

We have everything from Teacher Marked Essays to Mindmaps and Quizzes to help you with your work. Take a look around.

If you're stuck on how to get started, try creating some resources. It's free to do and can help breakdown tough topics into manageable chunks. Get creating now.

Thanks!

Not sure what all of this is about? Head here to find out more.

Reply 2

Silver Arrows

Don't know, but here's a Solar System mnemonic to help you remember the order of the planets

My Very Easy Method Just Speeds Up Naming Planets.

Spoiler

Reply 3

dknt

Original post by insparato

Hi, just been reading James Binney's Text on QM, came across this and didn't really understand his mathematical logic.

Let

< \vec{x} | \psi >

be a function of x

< \vec{x} | \psi> = \psi(x)

< \vec{x} - \vec{a}| \psi> = \psi(x) - \vec{a} \frac{\partial \psi}{\partial \vec{x}} + \frac{1}{2!} (\vec{a}.\frac{\partial \psi}{\partial \vec{x}})^2...

<\vec{x} - \vec{a}| \psi> = exp(-\vec{a} \frac{\partial}{\partial \vec{x}})\psi(x) = exp (-\vec{a}.\frac{\partial}{\partial \vec{x}}) < x | \psi >

Now this line I don't understand

= <x | exp (\frac{-i\vec{a}.\vec{p}}{\hbar})|\psi>

He mentions using

<\vec{x} | p | \psi > = -i\hbar \frac{\partial}{\partial \vec{x}} < \vec{x} | \psi>

I can't really see the connection, I sort see he may be substituting a rearrangement of the momentum operator for

\frac{\partial}{\partial x}

Cheers

All that's been done is as you've said i.e. substitute for the momentum operator. In 1D

\hat{p} = -i \hbar \frac{\partial}{\partial x} \implies \frac{\partial}{\partial x} = -\frac{1}{i \hbar} \hat{p} = \frac{i}{\hbar} \hat{p}

More generally, for various symmetries then we can construct an operator which evolves the system upon which it acts. In this case, the momentum operator is a generator for spatial symmetry. The Hamiltonian is the generator for symmetry in time. For most cases, the operator associated with evolving such a system is

\hat{O} = \exp(-\frac{i}{\hbar} [\mathrm{parameter}] \times \hat{G})

, where

\hat{G}

is the generator for the relevant symmetry. So for spatial evolution i.e. translation, the parameter is some vector a to displace by and the generator is momentum. For time evolution, it is some time t and the Hamiltonian etc.

(edited 9 years ago)

Reply 4

insparato

Cheers, don't know why I was getting myself in knots over that. Thanks for taking the time anyways.

I find his book frustrating and illuminating, it's unlike any other text in QM.