a) Show that Ψ0 are Ψ1 are orthogonal and that Ψ is normalized. b) Calculate the mean value of x and p in the states Ψ0, Ψ1 and Ψ.
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a) Show that Ψ0 are Ψ1 are orthogonal and that Ψ is normalized.
b) Calculate the mean value of x and p in the states Ψ0, Ψ1 and Ψ.
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- a) Write down the one-dimensional time-dependent Schro ̈dinger equation for a wavefunction Ψ(t, x) in a potential V (x). b) Write down the one-dimensional time-independent Schro ̈dinger equation for a wavefunc- tion ψ(x) in a potential V (x). c) Assuming that Ψ(t,x) corresponds to an energy eigenstate, write down a mathematical expression that relates the solutions of the one-dimensional time-dependent and time- independentSchro ̈dingerequations,Ψ(t,x)andψ(x).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. 2miii) Consider a 2D square potential energy well with sides L (length) containing six electrons. The potential energy is infinite at the sides and zero inside. The h? single-particle energies are given by 8mL +n), where n and ny are integers. If a seventh electron is added to the system when it is in its ground state find the least energy the additional electron can have?
- Taking the n=3 states as a representative example, explain the relationship between the complexity of hydrogen’s standing waves in the radial direction and their complexity in the angular direction at a given value of n. What relationship would this be considered a direct relationship or inverse relationship?7. Consider a particle in an infinite square well centered at x = 0 in one of its stationary states. For this problem, you may look up any integrals. Some useful ones are given in Harris. a) Compute (x) and (pr) for arbitrary n. Do this by direct computation but then describe how you could have found these results using symmetry (the symmetry can either be symmetry in the physical system, such as the shape of the wave function, or symmetry related to the expectation value integral, such as the shape of the integrand). b) Using your answer to part a), show that the uncertainty in the momentum is Apx nh for arbitrary n. Do this two ways: (i) first by using your answer to part a) and directly computating (p2) (via an integral) and (ii) by using your answer to part a) and relating (p2) to the kinetic energy operator. c) Show that the uncertainty principle holds for the ground state. 2L -24. Consider a modified box potential with V(x) = V₁x, Vi(ar), x a Use the orthogonal trial function = c₁f₁+c₂f₂ with f₁ = √√sin (H) and f2 = √√ √√sin sin (2) to determine the upper bound to ground state energy.