A particle in an infi nite square-well potential has ground-state energy 4.3 eV. (a) Calculate and sketch the energies of the next three levels, and (b) sketch the wave functions on top of the energy levels
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A particle in an infi nite square-well potential has ground-state energy 4.3 eV. (a) Calculate and sketch the energies of the next three levels, and (b) sketch the wave functions on top of the energy levels
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- An electron is trapped in a one-dimensional infinite potential well. For what (a) higher quantum number and (b) lower quantum number is the corresponding energy difference equal to the energy of the ng level? (c) Can a pair of adjacent levels have an energy difference equal to the energy of the n₂? (a) Number (b) Number i (c) Units UnitsElectron is confined in a 1D infinite potential well: U(x) = 0 at -a a. Using TIPT, calculate how the energy of the ground state is changed by a weak disturbance V = -Fr caused by a uniform electric field F.Consider a particle in the n = 1 state in a one-dimensional box of length a and infinite potential at the walls where the normalized wave function is given by 2 nTX a y(x) = sin (a) Calculate the probability for finding the particle between 2 and a. (Hint: It might help if you draw a picture of the box and sketch the probability density.)
- An electron is trapped inside a 1.00 nm potential well. Find the wavelength of the photons when the electron makes a transition from n =4 to n= 1.a) Show that if the total energy ε of a single particle state can be written as the sum of independent energies EiA, εiB, εic... then its partition function will factorise into a product of partition functions ZAZBZC. b) Given the factorisation, show how the free energy F and quantities such as S and Cy can be expressed as a sum of terms dependent on the sources A, B, C.15 6x4 – 12Lx³ + 15 L²x? - 3,3 (- L²r? L3x 6r4 - N2 2me 12La3 dx = dx 2 2 2 Evaluate the integral in the numerator. p*H@ dx = N² 40m h² L³ e 40me The denominator of the E expression from Step 1 is 6 _ 3Lx + 13 ,2 4 -L²x 4 Lx³ + F8-3Lz' + 12rt - (%) Lz3 + +L교2 p dx = N² dx Evaluate the integral in the denominator. 1 1 φ φ αχx = N2 840 840 Step 3 of 6 Divide the numerator by the denominator (both from Step 2) and simplify. (Use the following as necessary: ħ, L, me, P, T, and x.) 21h? E L²m 21h? L²me e Step 4 of 6 Calculate the energy for an electron in a 0.43-nm box using the formula from Step 3. = 4.0 0.26 X 840 kJ•mol-1 φ Step 5 of 6 Calculate the exact energy for an electron in the first excited state in a 0.43-nm box. n2h2 Recall that for a particle in a one-dimensional box we can write En we can therefore calculate an exact solution. 8mL2' Eexact = 4.0 197 X kJ-mol-1 Submit
- 4) Consider the one-dimensional wave function given below. (a) Draw a graph of the wave function for the region defined. (b) Determine the value of the normalization constant. (c) What is the probability of finding the particle between x = o and x = a? (d) Show that the wave function is a solution of the non-relativistic wave equation (Schrodinger equation) for a constant potential. (e) What is the energy of the wave function? (x) = A exp(-x/a) for x > o (x) = A exp(+x/a) for x < oa 4. 00, -Vo, V(z) = 16a 0, Use the WKB approximation to determine the minimum value that V must have in order for this potential to allow for a bound state.A particle is in the ground state of an infinite square well potential given by, 0 for -a sxsa V(x) = otherwise The probability to find the particle in the interval between a and 2 is