What fraction (as a percentage) does the n=(2×9-1)th infinite potential well wavefunction contribute to the 'classical' initial wavefunction ψ(x,t=0)=1/L1/2 ? (Why are the even n excluded?)
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What fraction (as a percentage) does the n=(2×9-1)th infinite potential well wavefunction contribute to the 'classical' initial wavefunction ψ(x,t=0)=1/L1/2 ? (Why are the even n excluded?)
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- Find the normalization constant B for the combination 18. As noted in Exercise 8, a linear combination of two wave functions for the same sysstem is also a valid wave function also a valid wave function functions for the same system 2TX = B sin TX +sin L. L. of the wave functions for then = 1 and n = 2 states od %3D particle in a box L wide. [A + CO]For quantum harmonic insulators Using A|0) = 0, where A is the operator of the descending ladder, look for 1. Wave function in domain x: V(x) = (x|0) 2. Wave function in the momentum domain: $(p) = (p|0)Complete the derivation of E = Taking the derivatives we find (Use the following as necessary: k₁, K₂ K3, and 4.) +- ( ²) (²) v² = SO - #2² - = 2m so the Schrödinger equation becomes (Use the following as necessary: K₁, K₂, K3, ħ, m and p.) 亢 2mm(K² +K ² + K² v k₁ = E = = EU The quantum numbers n, are related to k, by (Use the following as necessary: n, and L₁.) лħ n₂ π²h² 2m √2m h²²/0₁ 2m X + + by substituting the wave function (x, y, z) = A sin(kx) sin(k₂y) sin(kz) into - 13³3). X What is the origin of the three quantum numbers? O the Schrödinger equation O the Pauli exclusion principle O the uncertainty principle Ⓒthe three boundary conditions 2² 7²4 = E4. 2m
- (Requires integral calculus.) Imagine that a quanton's wavefunction at a given time is y(x) Ae-x/al, where A is an unspecified = constant and a = 35 nm . If we were to perform an experiment to locate the quanton at this time, what would be the probability (as a percent) of a result within ±0.47 a = ±16.45 nm of the origin? The probability is Note: Round the final answer to one decimal place. %.An electron is trapped in an infinitely deep one-dimensional well of width 10 nm. Initially, the electron occupies the n = 4 state. Calculate the photon energy required to excite the electron in the ground state to the first excited state.. (1) Find the kinetic, potential and total energies of the hydrogen atorn in the 2nd excited level.
- (a) A quantum dot can be modelled as an electron trapped in a cubic three-dimensional infinite square well. Calculate the wavelength of the electromagnetic radiation emitted when an electron makes a transition from the third lowest energy level, E3, to the lowest energy level, E₁, in such a well. Take the sides of the cubic box to be of length L = 3.2 x 10-8 m and the electron mass to be me = 9.11 x 10-³¹ kg. for each of the E₁ and E3 energy (b) Specify the degree of degeneracy levels, explaining your reasoning.An electron in a hydrogen atom is approximated by a one-dimensional infinite square well potential. The normalised wavefunction of an electron in a stationary state is defined as *(x) = √√ sin (""). L where n is the principal quantum number and L is the width of the potential. The width of the potential is L = 1 x 10-¹0 m. (a) Explain the meaning of the term normalised wavefunction and why normalisation is important. (b) Use the wavefunction defined above with n = 2 to determine the probability that an electron in the first excited state will be found in the range between x = 0 and x = 1 × 10-¹¹ m. Use an appropriate trigonometric identity to simplify your calculation. (c) Use the time-independent Schrödinger Equation and the wavefunction defined above to find the energies of the first two stationary states. You may assume that the electron is trapped in a potential defined as V(x) = 0 for 0≤x≤L ∞ for elsewhere.