EBK PHYSICS FOR SCIENTISTS & ENGINEERS
5th Edition
ISBN: 9780134296074
Author: GIANCOLI
Publisher: VST
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Chapter 38, Problem 23P
To determine
The proof that for a particle in a perfectly rigid box, the wavelength of the wave function for any state is the de Broglie wavelength.
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(10) i) Use the quantum mechanical kinetic energy operator T, =
to find the kinetic energy of the first
2m dx?
excited state of the Particle in a Box (with n = 2). ii) Then, use the relationship of kinetic energy and momentum (KE
= p?/2m) to find an equation for the de Broglie wavelength of the particle in a one-dimensional box as a function only
of the box length L and quantum number n. iii) Sketch the wavefunction in the box to verify that the expression you
obtained in part ii) is correct.
(c) Light of wavelength 450 nm is emitted by an electron in an atom behaving as a
lightly damped simple harmonic oscillator with a Q value of 2×107. Find the
natural frequency of the system in radians per second.
(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel
through the potential barrier of the width L, the difference between the barrier height U and the
incident energy E of the particle with mass m has to be close. Using the transmission
probability given in the text book / lecture, obtain the energy difference U-E which gives the
transmission probability of exp(-2).
(ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of
the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales
linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as
"shadows" of each other.
Energy
U
---E«
Electron
X
L
L
(iii) We now consider a 3-dimensional infinite square well potential having the length of the x, y, and z
directions to be all L. V=L**3 is the volume of the cube of this potential. We consider energy level of a
single particle (boson)…
Chapter 38 Solutions
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Similar questions
- (b) (deBroglie wave length) Determine the deBroglie wavelength (formula: p=h/2) of a grain of dust with diameter 1um, density 1kg m-3 and speed 1cm s. Compare your result with the diameter of the dust grain and the diameter of an atom. Comment?arrow_forward(i) We consider a one-dimensional potential barrier problem. In order for the particle to tunnel through the potential barrier of the width L, the difference between the barrier height U and the incident energy E of the particle with mass m has to be close. Using the transmission probability given in the text book / lecture, obtain the energy difference U-E which gives the transmission probability of exp(-2). (ii) We consider an infinite square well potential with the width L. Obtain the energy E_{gr} of the lowest energy level (ground state) of the particle with mass m, and show that E_{gr} scales linearly with E-U in the problem (i). The potential structures of (i) and (ii) can be viewed as "shadows" of each other. Energy U ---E- Electron X L Larrow_forward(ii) A beam of electrons is incident on a barrier 5 eV high and 0.5 nm wide. Find the energy they should have if 1% of them are to get through the barrier.arrow_forward
- Asaparrow_forwardAn electron is confined between two perfectly reflecting walls separated by the distance 12 x 10-11m. Use the Heisenberg uncertainty relation to estimate the lowest energy that the particle can have (in eV).arrow_forwardShow that the longest wavelength of the Balmer series and the longest two wavelengths of the Lyman series sat- isfy the Ritz combination principle. For the Lyman series, limit = 91.13 nm.arrow_forward
- 7) A particular laser oscillating at the resonant wavelength of 1 um has a M value of 1010. The photon number in the cavity at the threshold pumping rate is (Write the Answer as an integral number without using exponents)arrow_forwardIf in a box with infinite walls of size 2 nm there is an electron in the energy state n=1, find its (a) probability density, (b) the wave function and (c) the corresponding energy.arrow_forwardA quantum particle in an infinitely deep square well has a wave function that is given by ψ1(x) = √2/Lsin (πx/L)for 0 ≤ x ≤ L and is zero otherwise. (a) Determine the probability of finding the particle between x = 0 and x = 1/3L.(b) Use the result of this calculation and a symmetry argumentto find the probability of finding the particle between x = 1/3 L and x = 2/3 L. Do not re-evaluate the integral.arrow_forward
- 3) (a) Suppose, I have normalized the wave function at some point of time. The wave function evolves with time according to time dependent Schrodinger equation. How do I know that the wave function remains normalized after some time? [Hint: Show that –l4(x,t)|²dx = 0] dt (b) Show that d(p) %3D = (- dt ax i.e. expectation values follow Newton's law.arrow_forward(5) The wave function for a particle is given by: (x) = Ae/L for x 0, where A and L are constants, and L>0. (x) 0 for r <0. (a) Find the value of the constant A, as a function of L. A useful integral is: fe-K*dx =D-e-K, where K is a constant. (b) What is the probability of finding the particle in the range -10 Larrow_forwardConsider an electron, a proton, and an alpha particle (a helium nucleus), each trapped separately in identical boxes. (i) Which particle corresponds to the highest ground-state energy? Also which particle have longest wavelength when system is in ground statearrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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