An electron is confined in the ground state of a one-dimensional har- monic oscillator such that V((r – (x))²) = 10-10 m. Find the energy (in eV) required to excite it to its first excited state. (Hint: The virial theorem can help.]
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- The coherent states for the one-dimensional harmonic oscillator are defined as eigenstates of the operatorof annihilation a (which is non-Hermitian):a |λ⟩ = λ |λ⟩ (1)where λ is a complex number in general. a)prove that is a normalized consistent state. b)Show that the above state satisfies the minimum uncertainty relation, i.e., show thatWhat is the angle between the angular momentum vector L and the z-axis for a hydrogen atom in the state n=4, l=2, m=1? Give your numerical answer in degrees with one decimal place please.[q]Please do A, B, and C
- 2) Consider a particle in a three-dimensional harmonic oscillator potential V (r, y, z) = 5mw²(r² + y² + z®). The stationary states of such a system are given by ntm(r, y, z) = vn(x)¢r(y)v'm(2) (where the functions on the right are the single-particle harmonic oscillator stationary states) with energies Entm = hw(n +l+m+ ). Calculate the lifetime of the state 201.Please, I want to solve the question correctly, clearly and conciselyComplete 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