ok.. so i expanded that, and got ψ(x)+aψ'(x)+a^2*ψ''(x)/2!+ ...
But the LHS gives me:
e^(i*a*px)[ψ(x)]=e^(a*dψ/dx)=e^(-iak*ψ(x)), which is clearly not the left hand side. Am I interpreting the operator px wrong? This class is not a quantum dynamics class, so I'm having difficulty figuring...
Ok. so would the taylor series be:
ψ(a+a)+ψ'(a+a)(x-a)+ψ''(a+a)*(x-a)^2/2!+... ? and maybe disregard the higher order terms O(3)? Or would it be
ψ(a)+ψ'(a)*(x)+ψ''(a)*(x)^2/2!+... ? Either way, I'm not sure how I would prove that T(a) is the given exponential function..
Homework Statement
A translation operator T(a) coverts ψ(x) to ψ(x+a),
T(a)ψ(x) = ψ(x+a)
In terms of the (quantum mechanical) linear momentum operator p_x = -id/dx, show that T(a) = exp(iap_x), that is, p_x is the generator of translations. Hint. Expand ψ(x+a) as a Taylor series...
Homework Statement
"Determine whether the following series converge:
\sum_{n \geq 2} \frac{n^{ln (n)}}{ln(n)^{n}}
and
\sum_{n \geq 2} \frac{1}{(ln(n))^{ln(n)}}
Homework Equations
The convergence/divergence tests (EXCEPT INTEGRAL TEST):
Ratio
Dyadic
Comparison
P-test...