Concept explainers
Verify that the following wavefunctions are indeed eigenfunctions of the Schrödinger equation, and determine their energy eigenvalues.
(a)
(b)
(c)
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Physical Chemistry
- 5. Consider a particle constrained to move in one dimension described by the wavefunction v (x) = Ne2** (a) Determine the normalization constant (b) Is the wavefunction an eigenfunction of d? +16x? dx? (c) Calculate the probability of finding the particle anywhere along the negative x-axisarrow_forwardA normalized wavefunction for a particle confined between 0 and L in the x direction is ψ = (2/L)1/2 sin(πx/L). Suppose that L = 10.0 nm. Calculate the probability that the particle is (a) between x = 4.95 nm and 5.05 nm, (b) between x = 1.95 nm and 2.05 nm, (c) between x = 9.90 nm and 10.00 nm, (d) between x = 5.00 nm and 10.00 nm.arrow_forwardWhat are the allowed total angular momentum quantum numbers of a composite system in which j1 = 5 and j2 = 3?arrow_forward
- Consider the wave function V222(x, ỹ, z) for a particle in a cubic box. Figure 4.45a shows a contour plot in a cut plane at ž = 0.75. (a) Convince yourself that the contour plot in a cut at ž = 0.25 would have the same pattern, but each positive peak would become negative, and vice versa. (b) Describe the shape of this wave function in a plane cut at ỹ = 0.5.arrow_forward106. Combining two real wave functions ₁ and 2, the following functions are constructed: A = ₁ + $₂₂ B = = ₁ +i0₂, C = ₁ −i0₂, D=i(0₁ +0₂). The correct statement will then be (a) A and B represent the same state (c) A and D represents the same state (b) A and C represent the same state. (d) B and D represent the same state.arrow_forwardConsider a 1D particle in a box confined between a = 0 and x = 3. The Hamiltonian for the particle inside the box is simply given by Ĥ . Consider the following normalized wavefunction 2m dz² ¥(2) = 35 (x³ – 9x). Find the expectation value for the energy of the particle inside the box. Give your 5832 final answer for the expectation value in units of (NOTE: h, not hbar!). In your work, compare the expectation value to the lowest energy state of the 1D particle in a box and comment on how the expectation value you calculated for the wavefunction ¥(x) is an example of the variational principle.arrow_forward
- The ground state wave function for a particle in a one-dimensional box is of length L is y = (2/L)¹² sin(7x/L). Calculate the probability of the particle between x=4.00 nm to x = 4.80 nm. Assume the length of the box is 8.5 nm. Answer Choices: (A) 0.840 (B) 0.143 (C) 0.186 (D) 0.256arrow_forwardn = 9.19 (a) Calculate Ax Apx for a particle in a linear box for 1, 2, and 3 using the equation in Example 9.13. Compare these values with the minimum product of uncertainties from the Heisenberg uncertainty principle. (b) What is the uncer- tainty in x for a particle in a 0.2-nm box when its quantum num- ber is unity? In a 2-nm box?arrow_forwardThe normalized wave function for a particle in a one- dimensional box in which the potential energy is zero is (x) = /2/L sin (nTx/L), where L is the length of the box (with the left wall at x = 0). What is the probability that the particle will lie between x = 0 and x = ticle is in its n = 2 state? L/4 if the par-arrow_forward
- (a) What are the possible values for mℓ when the principal quantum number (n) is 2 and the angular momentum quantum number (ℓ) is 0? (b) What are the possible values for mℓ when the principal quantum number (n) is 3 and the angular momentum quantum number (ℓ) is 2?arrow_forward(a) What is the lowest possible value of the principal quantum number (n) when the angular momentum quantum number (ℓ) is 1? (b) What are the possible values of the angular momentum quantum number (ℓ) when the principal quantum number (n) is 4 and the magnetic quantum number (mℓ) is 0?arrow_forwardFor the system described in Exercise E7B.1(a) (A possible wavefunction for an electron in a region of length L (i.e. from x = 0 to x = L) is sin(2πx/L). Normalize this wavefunction (to 1)), what is the probability of finding the electron between x = L/4 and x = L/2?arrow_forward
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