- #1

FatPhysicsBoy

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## Homework Statement

I have quite a straightforward question on the taylor expansion however I will try to provide as much context to the problem as possible:

##T(a)## is unitary such that ##T(-a) = T(a)^{-1} = T(a)^{\dagger}## and operates on states in the position basis as ##T(a)|x\rangle = |x+a\rangle##

It has already been shown that ##\langle x | T(a) | \psi \rangle = \psi(x-a)## using the continuous form of the closure relation.

Now it must be shown that for an infinitesimal translation ##\delta a## such that ##\langle x | T(\delta a) | \psi \rangle = \psi(x-\delta a)##, via a taylor expansion of ##\psi(x-\delta a)## that the generator of infinitesimal translations acting on ##\psi(x)## takes the form:

##T(\delta a) = I - i\frac{\delta a}{\hbar}\tilde{p} + O(\delta a ^{2})##, where ##\tilde{p} = -i\hbar \frac{d}{dx}##

where ##I## is the identity operator.

## Homework Equations

1) ##f(x) = f(x_{0}) + \frac{df(x_0)}{dx}(x-x_0) + \frac{d^{2}f(x_0)}{dx^{2}}(x-x_0)^{2} + ..##

(This is what is used in the solutions, shouldn't there be a ##1/2!## factor in the 3rd term or is it just ignored for approximation purposes..?)

## The Attempt at a Solution

I am confused as to the motivations used in the solution to this problem where a parameter ##\Delta x = x - x_0## is defined and the taylor series above is recast in the following form:

2) ##f(x_0 + \Delta x) = f(x_0) + \frac{df(x_0)}{dx}(\Delta x) + O((\Delta x)^{2})##,

the identification is then made that in our case ##\Delta x = -\delta a## and therefore:

3) ##\psi(x - \delta a) = \psi(x) - \frac{d\psi(x)}{dx}(\delta a) + O((\delta a)^{2}) = \psi(x) - i\frac{\delta a}{\hbar}\tilde{p}\psi(x)##.

I understand the motivation to choose ##\Delta x = x - x_0## but only in the context of 'steering' the algebra towards the desired solution since we can observe that ##T(a)## increases in powers of ##\delta a##. I see how this then works algebraically, ##x = x_0 - \delta a## and so we recover 2) with ##\Delta x = - \delta a##.

Although it works algebraically, I'm just slightly confused by what's actually happening. My understanding of the taylor expansion only stretches as far as 'If we want to analyse ##f(x)## around ##x = a##, we expand in terms of derivatives of ##f(a)##' applying that to 2) doesn't make sense though 'If we want to analyse ##f(x_0 - \delta a)## around ##\delta a ## we expand in terms of derivatives of ##x_0##?!'

So that's the problem, I don't want to let the fact that the algebra worked out lobotomise my understanding of what's going on with the taylor expansion. I feel like I'm yet to understand something further..

Thanks in advance for any insights! :)