Suppose that a particle moves in x direction for the period (0 ? x ? 0.5), in the wave function \(\Psi(x) = Ae^{ix^2}\) Find: 1- Normalization constant A 2- Determine the expectation value \(\langle \hat{X} \rangle\), \(\langle \hat{P}_x \rangle\) and \(\langle \hat{T}_x \rangle\)
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A particle is represented (at time $t=0$ ) by the wave function $$ \Psi(x, 0)=\left\{\begin{array}{ll} A\left(a^{2}-x^{2}\right) . & \text { if }-a \leq x \leq+a \\ 0, & \text { otherwise } \end{array}\right. $$ (a) Determine the normalization constant $A$. (b) What is the expectation value of $x$ (at time $t=0$ )? (c) What is the expectation value of $p$ (at time $t=0$ )? (Note that you cannot get it from $p=m d\langle x\rangle / d t$. Why not?) (d) Find the expectation value of $x^{2}$. (e) Find the expectation value of $p^{2}$. (f) Find the uncertainty in $x\left(\sigma_{x}\right)$.
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Consider a particle whose wave function is given by ψ(x) = Ae^{-ax^2} (A) What is the value of A if this wave function is normalized? (B) What is the expectation value of x for this particle? Let's apply the normalization condition: ∫ |ψ|^2 dx = ∫ (Ae^{-ax^2})^2 dx = A^2 ∫ e^{-2ax^2} dx = 1 (B) To find the expectation value of x, we use the formula: ⟨x⟩ = ∫ ψ*xψ dx = ∫ (Ae^{-ax^2})x(Ae^{-ax^2}) dx = A^2 ∫ xe^{-2ax^2} dx = 0
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A particle with mass $m$ moving along the $x$ -axis and its quantum state is represented by the following wave function: $$ \Psi(x, t)=\left\{\begin{aligned} 0, & x<0 \\ A x e^{-\alpha x} e^{-i E t / \hbar}, & x \geq 0 \end{aligned}\right. $$ where $\alpha=2.0 \times 10^{10} \mathrm{m}^{-1} .$ (a) Find the normalization constant. (b) Find the probability that the particle can be found on the interval $0 \leq x \leq L$. (c) Find the expectation value of position. (d) Find the expectation value of kinetic energy.
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