Concept explainers
(a)
The uncertainty in momentum of electron in terms of
(a)
Answer to Problem 10P
The uncertainty in momentum is
Explanation of Solution
Write the equation for uncertainty principle.
Here,
Conclusion:
Rewrite the above relation by substituting
Rewrite the above relation in terms of
Therefore, the uncertainty in momentum is
(b)
The kinetic energy of electron.
(b)
Answer to Problem 10P
The kinetic energy of electron will be
Explanation of Solution
Write the equation for momentum.
Here,
Conclusion:
Rewrite the above equation by substituting
Therefore, the kinetic energy of electron will be
(c)
The total energy of electron in terms of
(c)
Answer to Problem 10P
The total energy of electron in terms of
Explanation of Solution
Write the equation for total energy of electron.
Here,
Write the equation for
Here,
Rewrite the equation for
Conclusion:
Rewrite the above equation by substituting
Therefore, the total energy of electron in terms of
(d)
The value of
(d)
Answer to Problem 10P
The value of
Explanation of Solution
Write the condition for the value of
Conclusion:
Rewrite the above equation by substituting for
Rewrite the above relation in terms of
The calculated value
Therefore, the value of
(e)
The resulting total energy.
(e)
Answer to Problem 10P
The resulting total energy is
Explanation of Solution
Write the equation for
Rewrite the above equation by substituting
Conclusion:
Substitute
Therefore, the resulting total energy is
(f)
States that the answers are in agreement the conclusions of Bohr theory.
(f)
Explanation of Solution
The uncertainty of momentum of electron calculated in part (a) is
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Chapter 29 Solutions
Principles of Physics: A Calculus-Based Text
- Determine the distance between the electron and proton in an atom if the potential energy UU of the electron is 11 eV (electronvolt, 1 eV =1.6×10−19=1.6×10−19 J). Give your answer in Angstrom (1 A = 10-10 m).arrow_forwardIf an excited state of an atom has a lifetime of 3.0××10^−7 s, what is the minimum error associated with the measurement of the energy of this state? ΔE=____ ×10^−28 Jarrow_forwardAn electron is revolving around a proton in a circular orbit of radius r. The proton is assumed to be stationary. The total energy of this system is p? 1 e? E 2m 4TE, r where p and m denote the momentum and mass of the electron, respectively. Take the radius r to be an estimate of the uncertainty in position Ar, and the uncertainty in momentum Ap to be an estimate of p. Suppose that ArAp = ħ when the system is in the ground state. Show that the ground state energy is given by 1 me4 e 8h? E1 Give the numerical value for E, in electronvolts. Discuss if your results are consistent with Bohr's model for the hydrogen atom.arrow_forward
- A quantum particle of mass m is placed in a one-dimensional box of length L. Assume the box is so small that the particle’s motion is relativistic and K = p2/2m is not valid. (a) Derive an expression for the kinetic energy levels of theparticle. (b) Assume the particle is an electron in a box of length L = 1.00 × 10-12 m. Find its lowest possible kinetic energy. (c) By what percent is the nonrelativistic equation in error?arrow_forwardWhat are the (a) energy, (b) magnitude of the momentum, and (c) wavelength of the photon emitted when a hydrogen atom undergoes a transition from a state with n = 4 to a state with n = 2? (a) Number 2.55 Units eV (b) Number 1.3617 Units kg-m/s or N-s (c) Number 4.865976353 Units This answer has no unitsarrow_forwardA certain atom remains in an excited state for about 51.7 ns before emitting a 2.15-eV photon and transitioning to the ground state. What is the uncertainty in the frequency of the photon in Hz?arrow_forward
- A) By what factor is the uncertainty of the electron's position(1.36×10-4 m) larger than the diameter of the hydrogen atom?(Assume the diameter of the hydrogen atom is 1.00×10-8 cm.) B) Use the Heisenberg uncertainty principle to calculate Δx for a ball (mass = 122 g, diameter = 8.50 cm) with Δv = 0.425 m/s. C) The uncertainty of the (above) ball's position is equal to what factor times the diameter of the ball?arrow_forwardQ:-3(a)-An electron moves in the x direction with a speed of 2.8 x 106 m/s. We can measure its speed to a precision of 2 %. With what precision can we simultaneously measure its x coordinate? (b)- Repeat the calculations of the previous example in the case of a pitched baseball (m = 0.135 kg) moving at a speed of (44.5 ms). Again assume that its speed can be measured to a precision of 2%arrow_forward3. An electron is revolving around a proton in a circular orbit of radius r. The proton is assumed to be stationary. The total energy of this system is 1 e? E = 2m 4T€, r where p and m denote the momentum and mass of the electron, respectively. Take the radius r to be an estimate of the uncertainty in position Ar, and the uncertainty in momentum Ap to be an estimate of p. Suppose that ArAp= ħ when the system is in the ground state. Show that the ground state energy is given by 1 те4 E, = - 2 8h2 Give the numerical value for Ej in electronvolts. Discuss if your results are consistent with Bohr's model for the hydrogen atom.arrow_forward
- W | File 70 Paste 14+1+13+1+12+|+11·10 ·9·1·8·1·7·1·6·1·5·1·4·1·3·1·2·1·1····1·1·20 Home Document1 - Microsoft Word (Product Activation Failed) Insert Page Layout References Review View T Calibri (Body) 14 T Α Α΄ B-B-S ## T AaBbCcDc AaBbCcDc AaBbC AaBbCc AaBl AaBbCcl BIU abe X, X² A ab T 트플 1 Normal No Spaci... Heading 1 Heading 2 Title Subtitle Font Paragraph G Styles ·2·1·1·····1·1·2·1·3·1·4·1·5·1· 6 · 1 · 7 · 1 · 8 · 1 ·9·1·10·1·11·1·12·1·13· |·14·1·15· |· · |·17· 1 · 18 · | I I I I I ATOMIC AND NUCLEAR PHYSICS PLEASE ANSWER ALL QUESTIONS The motion of two interacting particles (atoms or nuclei) can be described by the following radial Schrödigner equation d l(l 1) −2² [12 a (rªd) – (C,+¹) + V(r)]Re(k;r) = ERe(k;r). 2μ dr where Re(r) is the radial wave function, μ = 2, the reduced mass, V the interacting potential, my+m₂ E the total energy, and k the wave number, given by k = 2μE h² 2. Using Re(k,r) = u₂(k,r) kr show that the above Schrödigner equation reduces to l(l + 1)_24² ď² +…arrow_forwardIf a proton has an uncertainty in its velocity of 5.90 × 10⁻⁴ m/s, what is the uncertainty (in meters) in its position?arrow_forwardThe nucleus of a hydrogen atom is a single proton, which has a radius of about 1.1 × 10-15 m. The single electron in a hydrogen atom orbits the nucleus at a distance of 5.3 x 10-¹1 m. What is the ratio of the density of the hydrogen nucleus to the density of the complete hydrogen atom? Number i 1.12E+13 Units (no units)arrow_forward
- Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningModern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning