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- https://chem.libretexts.org/Courses/Colorado_State_University/Chem_474%3A_Physical_Chemistry_I_(Levinger)/Study_Sessions/Study_Session_5%3A_Even_and_Odd_FunctionsFrom the integral of f(x)=e−x2/2a2 over an even interval, e.g., −∞ to ∞, what can you conclude about the integral of an even function over an even interval compared to the i...From the integral of f(x)=e−x2/2a2 over an even interval, e.g., −∞ to ∞, what can you conclude about the integral of an even function over an even interval compared to the integral of the same function over one half that interval, e.g., −∞ to 0 or 0 to ∞? Extrapolating from this function, what can you say about the integral of an odd function over an even interval, such as from −∞ to ∞?
- https://chem.libretexts.org/Courses/Colorado_State_University/Chem_474%3A_Physical_Chemistry_I_(Levinger)/Group_Work_Activities/Group_Work_02%3A_Operators_and_EigenvaluesˆA(f(x)+g(x))=ˆAf(x)+ˆAg(x) (the operator is distributive) In an eigenvalue problem, an operator applied to a function is equivalent to a constant value multiplied times th...ˆA(f(x)+g(x))=ˆAf(x)+ˆAg(x) (the operator is distributive) In an eigenvalue problem, an operator applied to a function is equivalent to a constant value multiplied times the function, that is, In this equation, a is the eigenvalue; it is just a real, imaginary or complex numerical constant. For ˆA=ddx, can any mathematical function, g(x) serve as the eigenfunction of ˆA or are there examples of g(x) that would not work?
- https://chem.libretexts.org/Courses/Colorado_State_University/Chem_474%3A_Physical_Chemistry_I_(Levinger)/Study_Sessions/Study_Session_6%3A_Vectors_and_Spherical_Polar_CoordinatesWe often describe them by separating into their component parts, in terms of unit vectors along the x, y, and z directions, i.e., ˆi, ˆj, and ˆk, respectively, as shown belo...We often describe them by separating into their component parts, in terms of unit vectors along the x, y, and z directions, i.e., ˆi, ˆj, and ˆk, respectively, as shown below Use the figure below to describe the length of vector →r in terms of x,y, and z? What is expression used to evaluate the average value of cosθ over the surface of a sphere?
- https://chem.libretexts.org/Courses/University_of_California_Davis/UCD_Chem_110A%3A_Physical_Chemistry__I/UCD_Chem_110A%3A_Physical_Chemistry_I_(Larsen)/Worksheets/00%3A_Introduction_to_Complex_NumbersBasics operations of complex numbers and functions are introduced.
- https://chem.libretexts.org/Ancillary_Materials/Worksheets/Worksheets%3A_Physical_Chemistry/GroupWork_01%3A_Separation_of_VariablesThe "separation of variables" is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs...The "separation of variables" is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of an equation. We would like to separate the variables t and y so that all occurrences of t appear on the right-hand side, and all occurrences of y appears on the left and multiply dy/dt.
- https://chem.libretexts.org/Courses/Colorado_State_University/Chem_474%3A_Physical_Chemistry_I_(Levinger)/Study_Sessions/Study_Session_3%3A_Quantum_Concepts_IWhat are the implications of the Heisenberg uncertainty principle? Given the operators ˆx=x, and ˆpx=−iℏddx, do these operators commute with each other? If two linear ope...What are the implications of the Heisenberg uncertainty principle? Given the operators ˆx=x, and ˆpx=−iℏddx, do these operators commute with each other? If two linear operators ˆA and ˆB share the same eigenfunctions, will these two operators commute? How large does a particle need to be for it to behave clasically all the time? How large does a box need to be for a quantum mechanical particle to appear free?
- https://chem.libretexts.org/Ancillary_Materials/Worksheets/Worksheets%3A_Physical_Chemistry/Variation_Approximation_for_the_Particle_in_a_Box_(Worksheet)Here the relative error of the trial function are obvious; it is largest near the ends of the box where φ(y) is small and < 0.2% throughout most of the box. Use the following basis set for a...Here the relative error of the trial function are obvious; it is largest near the ends of the box where φ(y) is small and < 0.2% throughout most of the box. Use the following basis set for a linear variational treatment of the PIB: f n (y) = (y 2 -1) n , n=1,2,...,M. (a) Calculate the variational energy and wave function of the ground state for M=1,3,5. (b) Analyse the errors in these variational results.
- https://chem.libretexts.org/Ancillary_Materials/Worksheets/Worksheets%3A_Physical_Chemistry/Angular_Momentum_II_(Worksheet)\(\hat L^2=\hat L_x^2+\hat L_y^2+\hat L_z^2 = \hat L_x \hat L_x+\hat L_y \hat L_y+\hat L_z \hat L_z =\\ -\hbar^2 \left[ (y^2+z^2)\dfrac{\partial^2}{\partial x^2} + (x^2+z^2)\dfrac{\partial^2}{\partial...\(\hat L^2=\hat L_x^2+\hat L_y^2+\hat L_z^2 = \hat L_x \hat L_x+\hat L_y \hat L_y+\hat L_z \hat L_z =\\ -\hbar^2 \left[ (y^2+z^2)\dfrac{\partial^2}{\partial x^2} + (x^2+z^2)\dfrac{\partial^2}{\partial y^2} + (y^2+x^2)\dfrac{\partial^2}{\partial z^2}-2 \left(xy\dfrac{\partial^2}{\partial x \partial y}+xz\dfrac{\partial^2}{\partial x \partial z}+zy\dfrac{\partial^2}{\partial z \partial y}\right)-2 \left(x \dfrac{\partial }{\partial x}+y \dfrac{\partial }{\partial y}+z \dfrac{\partial }{\partial z…
- https://chem.libretexts.org/Ancillary_Materials/Worksheets/Worksheets%3A_Physical_Chemistry/Basic_Quantum_Concepts_II_(Worksheet)Show that the eigenfunctions of the 1D particle in a box are orthonormal. Show that a dipole operator of the form, ˆμ=μ0ˆx can lead to a transition between two levels in the 1D partic...Show that the eigenfunctions of the 1D particle in a box are orthonormal. Show that a dipole operator of the form, ˆμ=μ0ˆx can lead to a transition between two levels in the 1D particle in a box. Show that eigenfunctions of degenerate energy states of the 2D particle in a box are orthogonal. How deos a dipole operator of the form ˆμ=μ0ˆr=μ0,xˆx+μ0,yˆy affect eigenfunctions of degenerate energy states of the 2D particle in a box?
- https://chem.libretexts.org/Ancillary_Materials/Worksheets/Worksheets%3A_Physical_Chemistry/Probability_and_Statistics_(Worksheet)The uncertain nature of things in the quantum world make it necessary for us to use probability and statistics to describe the likelihood of finding a system in a certain state. In addition to the mea...The uncertain nature of things in the quantum world make it necessary for us to use probability and statistics to describe the likelihood of finding a system in a certain state. In addition to the mean or average, we can compute the second moment, ⟨x2⟩=∑nj=1x2jpj and with this, we define the variance σ2x=⟨(x−⟨x⟩)2⟩=n∑j=1(x−⟨x⟩)2pj
- https://chem.libretexts.org/Ancillary_Materials/Worksheets/Worksheets%3A_Physical_Chemistry/Particle_in_a_Box%3A_Probability_(Worksheet)Show that the eigenfunctions of the 1D particle in a box are orthonormal. Show that a dipole operator of the form, ˆμ=μ0ˆx can lead to a transition between two levels in the 1D partic...Show that the eigenfunctions of the 1D particle in a box are orthonormal. Show that a dipole operator of the form, ˆμ=μ0ˆx can lead to a transition between two levels in the 1D particle in a box. Show that eigenfunctions of degenerate energy states of the 2D particle in a box are orthogonal. How does a dipole operator of the form ˆμ=μ0ˆr=μ0,xˆx+μ0,yˆy affect eigenfunctions of degenerate energy states of the 2D particle in a box?
