Wave Function
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Given the Gaussian distribution
a) Determine the constant A;
b) Determine \( \langle x\rangle \), \( \langle x^{2}\rangle \) e σ;
c) Sketch the graph of ρ(x).
\[
\begin{gather}
\rho(x)=A\operatorname{e}^{-\lambda(x-a)^{2}}
\end{gather}
\]
where A, a and λ are constants.a) Determine the constant A;
b) Determine \( \langle x\rangle \), \( \langle x^{2}\rangle \) e σ;
c) Sketch the graph of ρ(x).
At t = 0, a particle is represented by the following wave function
a) Normalize Ψ (that is, determine A in terms of a and b);
b) Sketch the graph of Ψ(x, 0) as a function of x; c) What is the most likely position of the particle at t = 0?
d) What is the probability of finding the particle to the left of a? Check your result in the limit cases, b = a and b = 2a;
e) What is the expectation value of x?
\[
\begin{gather}
\Psi (x,0)=
\left\{\begin{array}{l}
\;\dfrac{Ax}{a}\,,&\text{if}\;0\leqslant x\leqslant a\\
\;\dfrac{A(b-x)}{(b-a)}\,,&\text{if}\;a\leqslant x\leqslant b\\
\;\;0\,,&\text{otherwise}
\end{array}
\right.
\end{gather}
\]
where A, a, and b are constantsa) Normalize Ψ (that is, determine A in terms of a and b);
b) Sketch the graph of Ψ(x, 0) as a function of x; c) What is the most likely position of the particle at t = 0?
d) What is the probability of finding the particle to the left of a? Check your result in the limit cases, b = a and b = 2a;
e) What is the expectation value of x?
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