Physical Chemistry I
Physical Chemistry I CHM 3410
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This 9 page Study Guide was uploaded by Angelo Cassin on Monday October 12, 2015. The Study Guide belongs to CHM 3410 at Florida Atlantic University taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/221666/chm-3410-florida-atlantic-university in Chemistry at Florida Atlantic University.
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Date Created: 10/12/15
An operator is a symbol that designates a process that will transform one function into another function For example An operator DX given as follows DX d dX Means a first derivative of function fX with respect to X should be taken If two or more operators are applied simultaneously to a function the operator immediately adjacent to the function will operate on the function first giving a new function For example If DX d dX and Dy ddX to the function fXy X3y2 DX Dy X3y2 DX 2 X3y 6 Xzy Or Dy DX X3y2 Dy 3 Xzyz 6 Xzy The result of the two operations is independent of the order in which the operators are applied then these operators are said to commute That is DX Dy fXy Dy DX fXy Or DX Dy DX Dy fXy 0 Here the brackets are called as commuter brackets and the expression in the bracket itself an operator called the commutator of the operators In general we can say that two operators A and B commute if and only if AB BA fX 0 QuantumMechanical Operators The Schrodinger equation of a particle of mass m confined to a thre dimensional region and with a potential energy Uxyz is given as follow h2 dzw dzw dzw UXYZl I 8W 872m dx2 dy2 dz2 The mathematical expression for w that satisfies this equation generally exist only for certain values of 8 and these values are the energies of the states of the system We can write the above equation as h2 d2 d2 d2 UXYZl I 8W 872m dx2 dy2 dz2 The expression in the curly bracket is an example of an quantum mechanical operator and is known as Hamiltonian operator and its symbol is H The A hat signifies it as a operator Thus the equation can be written as Hw 8w The function which satisfies such equations are called as eigenfunctions and the values of constants calculated from such equations are called as eigenvalues Eigen is a German word and it means characteristic Postulates of Quantum Mechanics The wave functions from the Schrodinger equation or eigen functions from the operator equation can be used to deduce the information about the systems in each of allowed states Thus we introduce a set of postulates that are expressed in terms of quantummechanical operators 1 The value of each physical property can be deduced by operating on eigenfunctions with the operator corresponding to that physical property or by solving Schrodinger equation 2 There are two basic operator which can be used to deduce physical properties a Position Operator is for one dimensional system and it is the position coordinate of the system Position Operator X X b The momemtum operator in X direction is Momentum operator p h 2 7 i d dX 3 There are two different situations arises when we a value of any physical property is obtained by working on eigen functions a When quantized values are obtained For a particular physical property the operator A is such that Aw aw Where a is the number of allowed values of physical property For example when the Hamiltonian operator is used Hw 2 8w yields the values for 8 Any measurement of the system would show that it had an energy equal to one of the 8 values b When average values are obtained For a another physical property the operator B is such that But 2 bw Where ltbgt is average value of physical property obtained as ltbgt le dT M1sz For eg in the particleonline problem the particle is not restricted to certain positions along the line but we can find out the average position of the particle using above expression A derived operator for kinetic energy Operators for the other physical properties can be deduced from the position and momentum operators Kinetic energy is given as KE 111022 mo2 2m momentum2 2m Thus the kinetic energy operator T is given as Kineticenergy operator T 12mh27iddxh2niddx h28n2md2dX2 For threedimensional system Kineticenergy operator T 112 8 72 m dzdxz dZdy2 dZdzz Rotation in a plane Angular Features Angular momentum is an important physical property of atoms and molecules Angular momentum values are used to characteristics the electronics states of atoms and some molecules and for the rotational states of the molecules of gas Angular Momentum Operators Similar to kinetic energy operators angular momentum operators can be deduced from the basic position operator and linear momentum operator The direction and the magnitude of the angular momentum can be described by a vector This vector is perpendicular to the plane that contains he radius vector and the linear momentum vector Refer figure 97 Refer 96 The magnitude of this vector since the radius and linear momentum vector are perpendicular to each other is equal to the product of the magnitudes of these two vectors This can be eXpressed in term of coordinates X and y of particle and the values of angular momentum components PK and Py Thus the contribution to the angular momentum along the 2 directions are XPy and ny Thus The angular momentum LZ X Py y Px Thus in order to get the value of angular momentum operator LZ We use position operator X X and y y an d linear momentum operators Px h 2 7 i d dX and Pyh21iddy Thus we get Angular momentum operator 2 L2 h 2 7 i X d dy y ddX The polar coordinates are more convenient than Cartesian coordinates to calculate the angular momentums Refer figure 98 Thus X r COSQ and 1X do r sino y Also y r sine and dyd0 rcoso x In addition the derivative df d9 of any function f with respect to a can be written in terms of X and y as follows df d0 df dX dX do df dy dy do but dXdQ y and dydo X Therefore df d0 df dX y df dy X dfdo X df dy y df dX Thus in general we can write dde Xddyyddx Therefore the angular momentum operator LZ can be written in terms of its polar coordinates Angular momentum operator 2 L2 h 2 Tl l 1 d0 Rotational Kinetic Energy Operator The kinetic energy can be expressed in terms of angular velocity 0 oa2nv 2nxu2nr Ur The moment of Inertia I mr2 Therefore we can write KE 12 mo2 12 m r2 1 r 2 12 1 032 10221 Thus the we can express KE operator TZ as Tz h28n21d2d20
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