Q#39.82
In a TV picture tube the accelerating voltage is 15.0 kV, and the electron beam passes through an aperture 0.50 mm in diameter to a screen 0.300 m away.
(a) Calculate the uncertainty in the component of the electron’s velocity perpendicular to the line between aperture and screen.
(b) What is the uncertainty in position of the point where the electrons strike the screen?
(c) Does this uncertainty affect the clarity of the picture significantly? (Use nonrelativistic expressions for the motion of the electrons. This is fairly accurate and is certainly adequate for obtaining an estimate of uncertainty effects.)
Answer:
Apply the Heisenberg Uncertainty Principle. Let the uncertainty product have its minimum possible value, so
$\Delta x \Delta p_x$ = $\frac{\hbar}{2}$
Take the direction of the electron beam to be the -direction x and the direction of motion perpendicular to the beam to be the -direction.
(a) $\Delta v_y=\frac{\Delta p_y}{m}=\frac{\hbar}{2m\Delta y}$
$\Delta v_y=\frac{1.055 \times 10^{-34} \ J.s}{2(9.11 \times 10^{-31} \ kg)(0.5 \times 10^{-3} \ m)}$ = 0.12 m/s
(b) The uncertainty Δr in the position of the point where the electrons strike the screen is
$\Delta r = \Delta v_yt$
$\Delta r = \frac{\Delta p_y}{m} \frac{x}{v_x}$
$\Delta r = \frac{\hbar}{2m \Delta y} \frac{x}{\sqrt{2K/m}}$
with K = qV = $1.602 \times 10^{-19} C \times 15.0 \times 10^3 V$ = $2.403 \times 10^{-15}$ J
So, $\Delta r = (0.12 \ m/s) \frac{0.300 \ m}{\sqrt{\frac{2(2.403 \times 10^{-15}) \ J}{9.11 \times 10^{-31} \ kg}}}$
$\Delta r = 4.78 \times 10^{-10} \ m$
(c) This is far too small to affect the clarity of the picture.
Q#39.83
The neutral pion ($\pi^0$) is an unstable particle produced in high-energy particle collisions. Its mass is about 264 times that of the electron, and it exists for an average lifetime $8.4 \times 10^{-17}$ s of before decaying into two gamma-ray photons. Using the relationship E = m$c^2$ between rest mass and energy, find the uncertainty in the mass of the particle and express it as a fraction of the mass.
Answer:
We use $\Delta E \Delta t \geq \frac{\hbar}{2}$.
Take the minimum uncertainty product, so $\Delta E = $\frac{\hbar}{2\Delta t}$
Given, $\Delta t = 8.4 \times 10^{-17} \ s$, m = 264$m_e$, so
$\Delta E = \frac{1.055 \times 10^{-34} \ J.s}{2(8.4 \times 10^{-17}) \ s}=6.28 \times 10^{-19} \ J$
and from $\Delta E = \Delta m c^2$ ⇒ $\Delta m = \frac{\Delta E}{c^2}$
$\Delta m = \frac{6.28 \times 10^{-19} \ J}{(2.998 \times 10^8 \ m/s)^2} = 7.0 \times 10^{-36} \ kg$
So, $\frac{\Delta m}{m} = \frac{7.0 \times 10^{-36} \ kg}{(264)(9.11 \times 10^{-31} \ kg)}$
$\frac{\Delta m}{m} = 2.9 \times 10^{-9}$
The fractional uncertainty in the mass is very small.
Q#39.84
Quantum Effects in Daily Life? A 1.25-mg insect flies through a 4.00-mm-diameter hole in an ordinary window screen. The thickness of the screen is 0.500 mm.
(a) What should be the approximate wavelength and speed of the insect for her to show wave behavior as she goes through the hole?
(b) At the speed found in part (a), how long would it take the insect to pass through the 0.500-mm thickness of the hole in the screen? Compare this time to the age of the universe (about 14 billion years). Would you expect to see “insect diffraction” in daily life?
Answer:
The insect behaves like a wave as it passes through the hole in the screen.
(a) For wave behavior to show up, the wavelength of the insect must be of the order of the diameter of the hole. The de Broglie wavelength is
$\lambda = \frac{h}{mv}$
The de Broglie wavelength of the insect must be of the order of the diameter of the hole in the screen, so, λ ≈ 4.00 mm. The de Broglie wavelength gives
$v = \frac{h}{m \lambda}=\frac{6.026 \times 10^{-34} \ J.s}{(1.25 \times 10^{-6} \ kg)(0.00400 \ m)}$
$v = 1.33 \times 10^{-25} \ m/s$
(b) $t = \frac{x}{v}= \frac{0.000500 \ m}{1.33 \times 10^{-25} \ m/s}$
$t = 3.77 \times 10^{21} \ s = 1.4 \times 10^{10}\ yr$
The universe is about 14 billion years old $1.4 \times 10^{10}$ yr so this time would be about 85,000 times the age of the universe.
Don’t expect to see a diffracting insect! Wave behavior of particles occurs only at the very small scale.
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