Work out the problem of the infinite square well in two dimensions. In this case, the...

Work out the problem of the infinite square well in two dimensions. In this case, the potential is zero inside a square of sides of length L, and infinite outside the square. Find the eigenfunctions and energy eigenvalues.

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genius_generous answered 2 years ago

A particle in the infinite square well (width a) starts out being equally likely to be...

A particle in the infinite square well (width a) starts out being equally likely to be found in the first and last third of the well and zero in the middle third. What is the initial (t=0) wave function? Find A and graph the initial wave function. What is the expectation value of x? Show your calculation for the expectation value of x, but then say why you could have just written down the answer. Will you ever find the...

An electron is in the ground state of an infinite square well. The energy of the...

An electron is in the ground state of an infinite square well. The energy of the ground state is E1 = 1.35 eV. (a) What wavelength of electromagnetic radiation would be needed to excite the electron to the n = 4 state? nm (b) What is the width of the square well? nm

Consider two non-interacting particles in an infinite square well. One is in a state ψm, the...

Consider two non-interacting particles in an infinite square well. One is in a state ψm, the other in a state ψn with n≠m. Let’s assume that ψm and ψn are the ground state and 1st excited state respectively and that the two particles are identical fermions. The well is of width 1Å. What is the probability of finding a particle in the 1st excited state in a region of width 0.01Å? Does this change if the particles are distinguishable?

Consider a particle in an infinite square well, but instead of having the well from 0...

Consider a particle in an infinite square well, but instead of having the well from 0 to L as we have done in the past, it is now centered at 0 and the walls are at x = −L/2 and x = L/2. (a) Draw the first four energy eigenstates of this well. (b) Write the eigenfunctions for each of these eigenstates. (c) What are the energy eigenvalues for this system? (d) Can you find a general expression for the...

A particle in an infinite one-dimensional square well is in the ground state with an energy...

A particle in an infinite one-dimensional square well is in the ground state with an energy of 2.23 eV. a) If the particle is an electron, what is the size of the box? b) How much energy must be added to the particle to reach the 3rd excited state (n = 4)? c) If the particle is a proton, what is the size of the box? d) For a proton, how does your answer b) change?

Consider a particle of mass ? in an infinite square well of width ?. Its wave...

Consider a particle of mass ? in an infinite square well of width ?. Its wave function at time t = 0 is a superposition of the third and fourth energy eigenstates as follows: ? (?, 0) = ? 3i?3(?)+ ?4(?) (Find A by normalizing ?(?, 0).) (Find ?(?, ?).) Find energy expectation value, <E> at time ? = 0. You should not need to evaluate any integrals. Is <E> time dependent? Use qualitative reasoning to justify. If you measure...

Find the lowest two "threefold degenerate excited states" of the three-dimensional infinite square well potential for...

Find the lowest two "threefold degenerate excited states" of the three-dimensional infinite square well potential for a cubical "box." Express your answers in terms of the three quantum numbers (n1, n2, n3). Express the energy of the two excited degenerate states that you found as a multiple of the ground state (1,1,1) energy. Would these degeneracies be "broken" if the box was not cubical? Explain your answer with an example!

in this problem we are interested in the time-evolution of the states in the infinite square...

in this problem we are interested in the time-evolution of the states in the infinite square potential well. The time-independent stationary state wave functions are denoted as ψn(x) (n = 1, 2, . . .). (a) We know that the probability distribution for the particle in a stationary state is time-independent. Let us now prepare, at time t = 0, our system in a non-stationary state Ψ(x, 0) = (1/√( 2)) (ψ1(x) + ψ2(x)). Study the time-evolution of the probability...

Find the energy spectrum of a particle in the infinite square well, with potential U(x) →...

Find the energy spectrum of a particle in the infinite square well, with potential U(x) → ∞ for |x| > L and U(x) = αδ(x) for |x| < L. Demonstrate that in the limit α ≫ hbar^2/mL, the low energy part of the spectrum consists of a set of closely-positioned pairs of energy levels for α > 0. What is the structure of energy spectrum for α < 0?

Derive the general wavefunction for a particle in a box (i.e. the infinite square well potential)....

Derive the general wavefunction for a particle in a box (i.e. the infinite square well potential). Go on to normalise it. What energy/energies must the particle have to exist in this box?

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