Concept explainers
A particle on a ring has a wavefunction
where
How does the angular momentum depend on the constant
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Physical Chemistry
- A particle freely moving in one dimension x with 0 ≤ x ≤ ∞ is in a state described by the normalized wavefunction ψ(x) = a1/2e–ax/2, where a is a constant. Evaluate the expectation value of the position operator.arrow_forwardFor the system described in Exercise E7C.8(a), evaluate the expectation value of the angular momentum represented by the operator(ħ/i)d/dϕ for the case ml = +1, and then for the general case of integer ml.arrow_forwardThe rotation of a molecule can be represented by the motion of a particle moving over the surface of a sphere with angular momentum quantum number l = 2. Calculate the magnitude of its angular momentum and the possible components of the angular momentum along the z-axis. Express your results as multiples of ℏ.arrow_forward
- Calculate the probability that a particle will be found between 0.49L and 0.51L in a box of length L for (i) ψ1, (ii) ψ2. You may assume that the wavefunction is constant in this range, so the probability is ψ2δx.arrow_forwardNormalize (to 1) the wavefunction e-ax^2 in the range −∞ ≤ x ≤ ∞, with a > 0. Refer to the Resource section for the necessary integral.arrow_forwardBy considering the integral ∫02π ψ*ml ψml dϕ, where ml≠m'l, confirm that wavefunctions for a particle in a ring with different values of the quantum number ml are mutually orthogonal.arrow_forward
- The wave function for the ground state of the harmonic oscillator is Vo(x) = Ce-[mw/(2ħ)]x² where C is an arbitrary constant, ħ is Planck's constant divided by 2π, m is the mass of the particle, W = ✓k/m, and k is the "spring constant" for the harmonic oscillator. Part A Normalize this wave function. What is the (positive) value of C once this wave function is normalized? You will need the formula Se -∞ Express your answer in terms of w, m, ħ, and T. ► View Available Hint(s) C = 17 ΑΣΦ xa Xh عات a √x vx 18 X> IXI -ax² X.10n X = ? wwwwwwwwww √. aarrow_forwardThe rotation of a molecule can be represented by the motion of a particle moving over the surface of a sphere. Calculate the magnitude of its angular momentum when l = 1 and the possible components of the angular momentum along the z-axis. Express your results as multiples of ℏ.arrow_forwardA particle rotating on the surface of a sphere with fixed r has the quantum numbers I = 3 and mj = 1. What is the magnitude of the angular momentum? V12 h O 12n? O 12harrow_forward
- P7B.8 A normalized wavefunction for a particle confined between 0 and L in the x direction, and between 0 and L in the y direction (that is, to a square of side L) is y = (2/L) sin(Tx/ L) sin(Ty/L). The probability of finding the particle between x, and x, along x, and between y, and y, along y is P= "w°dxdy Calculate the probability that the particle is: (a) between x = 0 and x = L/2, y = O and y = L/2 (i.e. in the bottom left-hand quarter of the square); (b) between x = L/4 and x = 3L/4, y = L/4 and y = 3L/4 (i.e. a square of side L/2 centred on x = y = L/2).arrow_forwardConsider the three spherical harmonics (a) Y0,0, (b) Y2,–1, and (c) Y3,+3. (a) For each spherical harmonic, substitute the explicit form of the function taken from Table 7F.1 into the left-hand side of eqn 7F.8 (the Schrödinger equation for a particle on a sphere) and confirm that the function is a solution of the equation; give the corresponding eigenvalue (the energy) and show that it agrees with eqn 7F.10. (b) Likewise, show that each spherical harmonic is an eigenfunction of lˆz = (ℏ/i)(d/dϕ) and give the eigenvalue in each case.arrow_forwardConsider again the system in quizzes 1 and 2, namely a particle moving in one dimension described by the normalized wavefunction (x) = 30 1 (а — х) for 0 a . а Determine the expectation value () for the particle.arrow_forward
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