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- The wavefunction of is Ψ(x) = Axe(−ax^2)/2 for with energy E = 3aℏ2/2m. Find the bounding potential V(x). Looking at the potential’s form, can you write down the two energy levels that are immediately above E?arrow_forwardThe normalised wavefunction for an electron in an infinite 1D potential well of length 80 pm can be written:ψ=(0.587 ψ2)+(0.277 i ψ7)+(g ψ6). As the individual wavefunctions are orthonormal, use your knowledge to work out |g|, and hence find the expectation value for the energy of the particle, in eV.arrow_forwardWhat is the solution of the time-dependent Schrödinger equation (x, t) for the total energy eigenfunction 4(x) = √2/a sin(3mx/a) for an electron in a one-dimensional box of length 1.00 x 10-10 m? Write explicitly in terms of the parameters of the problem. Give numerical values for the angular frequency and the wave- length of the particle.arrow_forward
- The wavefunction for v =1 for a simple harmonic oscillator is Ψ = (2)1/2 ( α3/π)1/4 x exp (-αx2/2) Find the values of x such that ψ* ψ is a maximum.Hint: Differentiate dψ*ψ/dx and set the result equal to zero and solve for the value of x.arrow_forwardA particle of mass 1.60 x 10-28 kg is confined to a one-dimensional box of length 1.90 x 10-10 m. For n = 1, answer the following. (a) What is the wavelength (in m) of the wave function for the particle? m (b) What is its ground-state energy (in eV)? eV (c) What If? Suppose there is a second box. What would be the length L (in m) for this box if the energy for a particle in the n = 5 state of this box is the same as the ground-state energy found for the first box in part (b)? m (d) What would be the wavelength (in m) of the wave function for the particle in that case? marrow_forwardThe wavefunction of is Ψ(x) = Axe−αx2/2 for with energy E = 3αℏ2/2m. Find the bounding potential V(x). Looking at the potential’s form, can you write down the two energy levels that are immediately above ??arrow_forward
- A particle confined in a one-dimensional box of length L(0<=x<=L) is in a state described by the wavefunction Ψ(x)= Ψ1+ Ψ2(x). Where A and B are constants given by real numbers and A>=0.(With X=6 and Y=3)l)Determine what relationship A and B must satisfy for the wavefunction to be normalized.ll)Suppose that A=B .What is the probability of the particle being found in the interval 0<=x<=L/2?lll)What values of A and B that minimize the probability of finding the particle in the range of positions 0<=x<=L/2?arrow_forwardProblem 1: (a) A non-relativistic, free particle of mass m is bouncing back and forth between two perfectly reflecting walls separated by a distance L. Imagine that the two oppositely directed matter waves associated with this particle interfere to create a standing wave with a node at each of the walls. Find the kinetic energies of the ground state (first harmonic, n = 1) and first excited state (second harmonic, n = 2). Find the formula for the kinetic energy of the n-th harmonic. (b) If an electron and a proton have the same non-relativistic kinetic energy, which particle has the larger de Broglie wavelength? (c) Find the de Broglie wavelength of an electron that is accelerated from rest through a small potential difference V. (d) If a free electron has a de Broglie wavelength equal to the diameter of Bohr's model of the hydrogen atom (twice the Bohr radius), how does its kinetic energy compare to the ground-state energy of an electron bound to a Bohr model hydrogen atom?arrow_forwarda) Write down the one-dimensional time-dependent Schro ̈dinger equation for a particle of mass m described by a wavefunction Ψ(t, x) in a potential V (x). b) For energy eigenstates, the wavefunction can be written as Ψ(t, x) = f (t) ψ(x). For this wavefunction:(i) state the time-independent equation that must be satisfied by ψ(x). (ii) derive an expression for f(t), in terms of the energy of the particle E.arrow_forward
- Consider the function v(1,2) =( [1s(1) 3s(2) + 3s(1) 1s(2)] [x(1) B(2) + B(1) a(2)] Which of the following statements is incorrect concerning p(1,2) ? a. W(1,2) is normalized. Ob. The function W(1,2) is symmetric with respect to the exchange of the space and the spin coordinates of the two electrons. OC. y(1,2) is an eigenfunction of the reference (or zero-order) Hamiltonian (in which the electron-electron repulsion term is ignored) of Li with eigenvalue = -5 hartree. d. The function y(1,2) is an acceptable wave function to describe the properties of one of the excited states of Lit. Oe. The function 4(1,2) is an eigenfunction of the operator S,(1,2) = S;(1) + S,(2) with eigenvalue zero.arrow_forwardShow that ? (x,t) = A exp [i (kx - ?t] is a solution to the time-dependent Schroedinger equation for a free particle. What relationship must k and ? have?arrow_forwardA particle of mass m is trapped in a three-dimensional rectangular potential well with sides of length L, L/ √2, and 2L. Inside the box V = 0, outside V = ∞. Assume that Ψ = Asin (k1x) sin (k2y) sin (k3z) inside the well. Substitute this wave function into the Schrödinger equation and apply appropriate boundary conditions to find the allowed energy levels. Find the energy of the ground state and first four excited levels. Which of these levels are degenerate?arrow_forward
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