Chapter 1: Q17P (page 22)

A particle is represented (at time t=0) by the wave function
$Ψ (x) = {A (a^{2} - x^{2})}$ ; if $- a \leq x \leq + a$
$Ψ (x) = 0$ ; Otherwise
a) Determine the normalization constantA?
b) What is the expectation value of p(at time t=0)?
c) What is the expectation value of x(at time t=0)?
d) Find the expectation value of $x^{2}$ .
e) Find the expectation value of $p^{2}$ .
f) Find the uncertainty inrole="math" localid="1658551318238" $x (σ_{x})$ .
g) Find the uncertainty in $p (σ_{x})$ .
h) Check that your results are consistent with the uncertainty principle.

Short Answer

Expert verified

(a) $A = \sqrt{\frac{15}{16 a^{2}}}$ ,

(b) $(x) = 0$

(d) $X^{2} = \sqrt{\frac{a^{2}}{7}}$

(e) $p^{2} = \sqrt{\frac{5 h^{2}}{2 a^{2}}}$

(f) $σ_{x} = \sqrt{\frac{a^{2}}{7}}$

(g) role="math" localid="1658551495957" $σ_{y} = \sqrt{\frac{5 h^{2}}{2 a^{2}}}$

(h) It’s Consistent

Step by step solution

Step 1: Define the Schrödinger equation

A differential equation that describes matter in quantum mechanics in terms of the wave-like properties of particles in a field. Its answer is related to a particle's probability density in space and time.

Time-dependent Schrödinger equation is represented as

$ih \frac{d}{dt} | ψ (t) | \equiv \overset{⏜}{H} | ψ (t) |$

Calculation

$(a) 1 = \int_{- \infty}^{\infty} {[ψ (x, 0)]}^{2} d x = \int_{- a}^{a} A^{2} {(a^{2} - x^{2})}^{2} d x 1 = A^{2} \int_{- a}^{a} (a^{4} - 2 a^{2} x^{4}) = A^{2} {[a^{4} x - \frac{2}{3} a^{2} x^{3} + \frac{x^{5}}{5}]}_{- a}^{a} 1 = A^{2} \frac{16}{15} a^{2} A = \sqrt{\frac{15}{16 a^{2}}}$

(b)

$(x) = \int_{- \infty}^{\infty} ψ^{*} x ψ d x = \int_{- a}^{a} A (a^{2} - x^{2}) x A (a^{2} - x^{2}) d x = 0$

(c)

$(p) = \int_{- \infty}^{\infty} ψ^{*} p ψ d x = - i h A^{2} \int_{- a}^{a} (a^{2} - x^{2}) \frac{\partial}{\partial x} (a^{2} - x^{2}) d x = - i h A^{2} \int_{- a}^{a} (2 a^{2} x - 2 x^{3}) = 0$

(d)

$X^{2} = \int_{- \infty}^{\infty} ψ^{*} x^{2} ψ d x = \int_{- a}^{a} A (a^{2} - x^{2}) x^{2} A (a^{2} - x^{2}) d x = A^{2} \int_{- a}^{a} (x^{2} a^{4} - 2 a^{2} x^{4} + x^{6}) d x = \frac{a^{2}}{7}$

(e)

p^{2} = \int_{- \infty}^{\infty} ψ^{*} p^{2} ψ d x = - 4 h^{2} A^{2} \int_{0}^{a} (a^{2} - x^{2}) d x = - 4 h^{2} A^{2} {[a^{2} x - \frac{x^{3}}{3}]}_{0}^{a} = \frac{5 h^{2}}{2 a^{2}}

(f)

$σ_{x} = \sqrt{x^{2} - x} = \sqrt{\frac{a^{2}}{7}}$

(g)

$σ_{p} = \sqrt{p^{2} - p} = \sqrt{\frac{5 h^{2}}{2 a^{2}}}$

(h)

$σ_{x} σ_{p} = \sqrt{\frac{a^{2}}{7}} \sqrt{\frac{5 h^{2}}{2 a^{2}}} = \sqrt{\frac{10}{7}} \frac{h}{2} > \frac{h}{2}$

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Short Answer

Step by step solution

Step 1: Define the Schrödinger equation

Calculation

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