CHEM 242 Chapter 2.pdf

Dr. Soha Albukhari 1 QUANTUM CHEMISTRY AND STATISTICAL THERMODYNAMICS CHEM 242 Chapter 2 The Postulates and General Principles of Quantum Mechanics Dr. Soha Albukhari [email protected] Chapter 2: The Postulates and General Principles of Quantum Mechanics 2 2.1 First Postulate: State of the System and Wave function (The state of a system is completely specified by its wave function) 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 2.3 Third Postulate: Absolute Values of Sharp Observables (Observable quantities must be eigenvalues of quantum mechanical operators) 2.4 Fourth Postulate: Expectation Values of Non-Sharp Observables 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation. Dr. Soha Albukhari Chapter 2: The Postulates and General Principles of Quantum Mechanics 3 ¨ Up to now we have made number of conjectures concerning the formulation of quantum mechanics. ¨ For example, we have been led to suspect that the variables of classical mechanics are represented in quantum mechanics by operators. These operate on wave functions to give the average or expected results of measurements. ¨ In this chapter we shall formalize the various conjectures as a set of postulates and then discuss some general theorems that follow from these postulates. Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 4 ¨ Classical mechanics deals with quantities called dynamical variables, such as position, momentum, angular momentum, and energy. A measurable dynamical variable is called an observable. ¨ The classical-mechanical state of a one-body system at any particular time is specified completely by the three position coordinates (x , y , z) and the three momenta or velocities ( vx , vy , vz ) at that time. ¨ The time evolution of the system is governed by Newton's equations Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 5 Newton’s equations m(d2x/dt2)=Fx , m(d2y/dt2)=Fy , m(d2z/dt2)=Fz where Fx , Fy and Fz are the components of the force F(x, y , z). Realize that generally each force component depends upon x, y, and z. To emphasize this, we write: m(d2x/dt2)=Fx(x, y, z) m(d2y/dt2)=Fy(x, y, z) m(d2z/dt2)=Fz(x, y, z) Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 6 ¨ Each of these equations is a second-order equation, there will be two integration constants from each one, the initial positions and initial velocities and write them as x0, y0, z0, vx0, vy0, and vz0. ¨ The solutions to pervious equation are x(t) , y(t), and z(t) , which describe the position of the particle as a function of time. ¨ The position of the particle depends not only on the time but also on the initial conditions. To emphasize this, we write the solutions to be: x(t)= x(t; x0, y0, z0, vx0, vy0, vz0) y(t)= y(t; x0, y0, z0, vx0, vy0, vz0) z(t)= z(t; x0, y0, z0, vx0, vy0, vz0) Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 7 ¨ We can write these three equations in vector notation: r(t)=r(t; r0, v0) ¨ The vector r(t) describes the position of the particle as a function of time; r(t) is called the trajectory of the particle. Classical mechanics provides a method for calculating the trajectory of a particle in terms of the forces acting upon the particle through Newton’s equations. Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 8 ¨ Thus, we see that in classical mechanics we can specify the state of a system by giving 3N positions and 3N velocities or momenta. ¨ We should suspect immediately that this is not going to be so in quantum mechanics because the uncertainty principle tells us that we cannot specify or determine the position and momentum of a particle simultaneously with infinite precision. ¨ The uncertainty principle is not important for macroscopic bodies; however, classical mechanics is a perfectly adequate prescription for macroscopic bodies; but for small bodies such as electrons, atoms, and molecules, the consequences of the uncertainty principle are far from negligible and so the classical-mechanical picture is not valid. ¨ This leads us to our first postulate of quantum mechanics. Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 9 First Postulate: The state of a quantum-mechanical system is completely specified by a function Ψ(r, t) that depends on the coordinates of the particle and on time. This function, called the wave function or state function, has the important property that Ψ*(r, t)Ψ(r, t) dxdydz is the probability that the particle lies in the volume element dxdydz located at (r) and at time (t). If there is more than one particle, say two, then we write: Ψ*(r1, r2, t)Ψ(r1, r2, t) dx1dy1dz1dx2dy2dz2 Ψ (ebsi) Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 10 ¨ For the probability that particle 1 lies in the volume element dx1dy1dz1 at r1 and that particle 2 lies in the volume element dx2dy2dz2 at r2 at time t. Postulate 1 says that the state of a quantum-mechanical system such as two electrons is completely specified by this function and that nothing else is required!. ¨ Because the square of the wave function has a probabilistic interpretation, it must satisfy certain physical requirements. For example, a wave function must be normalized, so that in the case of one particle, for simplicity, we have: ∞ ∫ −∞ dτΨ*(r, t)Ψ(r, t) = 1 , dτ = dxdydz Normalized wave function Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 11 Normalization Condition: ¨ Suppose that the system is an electron moving in the x direction. ¨ The state function (Ψ) describes the state of this electron as it moves in the x-direction ¨ The square of the state function at the region dx (Ψ2) → probability of finding it at this region. a dx b x direction Represents the probability of finding the y dx2 electron in the region dx Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 12 ¨ dx: is a small region (region: x → x + dx) ¨ If we divide the whole distance between the two points a & b into small regions each called dx. ¨ If we calculate the probability in each region, the summation of all the probabilities in all regions should equal to unity (= 1). a dx dx dx dx dx dx b x direction Ψ2dx Probability: ( 20% 10% 30% 25% 5% 10% ) = 100% =1 b òy dx = 1 2 Dr. Soha Albukhari a 2.1 First Postulate: The state of a system is completely specified by its wave function 13 ¨ For an electron moving in all directions (x, y, z): ∫ ψ 2 dx dy dz = 1 all space ∫ ψ 2 dτ = 1 d τ = dx dy dz all space ¨ Normalization Condition means that the wave function is a function normalized exactly all over the available space that the system can spread in. ¨ This function is called Normalized Wave function. Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 14 Orthogonality Condition ¨ Each system has many states, each state could be fully described by a wave function Ψ. State 3 Ψ3 State 2 ¨ Ψ1 is a state function of state ‘’1’’ that Ψ2 State 1 contains all the information about the system Ψ1 while it is in state 1. ¨ Ψ1 has no information about the system while it is in state 2. State 2 is fully described by a state function Ψ2 ¨ Ψ1 is completely independent on Ψ2 Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 15 ¨ This independency of the wave functions could be expressed mathematically as: òy y all space 1 2 dt = 0 ¨ Generally: òy y all space i j dt = 0 i¹ j ¨ This equation is called the Orthogonality Condition: represents the independence of all Ψ’s on each other. Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 16 Each wave function Ψi is Each Pair of wave functions are Normalized Orthogonal òy y dt = 1 òy y dt = 0 * i i i j i¹ j all space all space The Wave function is Orthonormal òy y all space i j dt = d ij d ij = 0 i ¹ j =1 i = j Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 17 Important: ¨ Electron motion in an atom or molecule can be described by a function called the wave function denoted as Ψ (ebsi) (the wave motion of the confined electron). ¨ The wave function of the electron depends on the position of the electron (Cartesian coordinates x, y, and z) and on the time (t). Ψ(x, y, z, t) Ψ(r, t) Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 18 ¨ Ψ(r, t) Must be well-Behaved wave function and follow these conditions: 1. Single-Valued: The function has only one value for each value of the changeable values that it depends on. F(r) has three values at r1 F(r) F(r) is not a single-values function r1 r F(r) Not well-behaved 1. Continuous: The function cover all the states available to the system 2. Finite: The function should not equal to infinity at any point in space. Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 19 ¨ The well-behaved wave function conditions implies that the square of the wave function of the system at any point gives the Probability of finding the system at this point. IΨ(r)I2 = Probability of e- ¨ In classical mechanics probability does not exist? (where is the car?) ¨ In quantum mechanics we deal with wave! The wave is distributed all over the available space, if we ask about the probability means that the point at which the wave is more likely or less likely to be distributed! (What is the probability of finding the object at a specific point?) Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 20 ¨ De Broglie clarify that the electron has wave-nature, regarding to his hypothesis the system should: 1. Be infinitesimal small particle. 2. Moving with speed comparable to the speed of light. 3. Confined to a small space (atomic or molecular size!) ¨ The electron in an atom is a system that could be fully explained using quantum mechanics. ¨ Also, De Broglie clarify that the electron can occupy only a definite states (allowed states). Therefor, the electron is a system of definite states. Dr. Soha Albukhari 2.1 First Postulate: The state of a system is completely specified by its wave function 21 ¨ State Function (Wave Function): It is a Mathematical abstraction for the state of the system. ¨ Stationary State: If the properties of the system in one of its state do not change with time, the system is said to be in a stationary state. ¨ Stationary State Wave Function: is the wave function of a stationary state. ¨ The state function by itself has no meaning. It is only a mathematical abstraction of the system state, it needs OPERATOR? (Second Postulate) Dr. Soha Albukhari 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 22 Mathematical Preliminaries 1- The complex function or number: is any quantity contains the imaginary number i , i = -1 c = a + ib is an complex number, y = eix is an complex function Each number or function has a Complex Conjugate * by replacing each i by -i c* = a – ib is the Complex Conjugate of for the number c. y* = e-ix is the Complex Conjugate of the function y Dr. Soha Albukhari 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 23 The value of i is imaginary that could not be determined. To get rid of i we use the Complex Conjugate as follows: ¨ The algebraic sum of any complex number and its conjugate is a real number: c + c* = (a + ib) + (a – ib) = 2a ¨ The product of multiplying the complex number by its conjugate is a real number: c. c* = (a + ib). (a – ib) = a2 + b2 ¨ The absolute value of c is given by: c = (cc * ) = a 2 + b 2 ¨ The probability is a real value. Therefor, If the state function is a complex function, then its square is also complex. Ψ* is complex conjugate. ò ydt = 1 y * all space Dr. Soha Albukhari 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 24 We concluded that classical-mechanical quantities are represented by linear operators in quantum mechanics. We now formalize this conclusion by our next postulate. Second Postulate: To every observable in classical mechanics there corresponds a linear operator (Hermitian operator) in quantum mechanics. The numerical value of this property is deduced from the mathematical properties of its quantum operator. Observable: any property that can be measured (e.g. total energy, kinetic energy, potential energy, momentum, ……..) Property = Observable Dr. Soha Albukhari 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 25 ¨ Classical mechanics operator: is a mathematical description indicate a specific mathematical process that should be performed on the expression that follow this operator. 2 d 2 dx [ x + 5x + 1 ] is the Mathematical Operator is the Mathematical Operator that that tells us to take the square indicate that we should perform a root for the number 2 differentiation with respect to x on expression follows [x2 + 5x + 1] Dr. Soha Albukhari 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 26 ¨ Quantum mechanics operator: The operator by itself has no specific value. We will denote the mathematical operator by the sign ^ (hat) placed above a letter. â , Pˆ , Qˆ ¨ The operator could be Real or Complex (contains the imaginary quantity) i = -1 ¨ The operator could be a Vector Operator (example: Del Ñ ) ¶ 2 ¶ 2 ¶ 2 Ñ2 = 2 + 2 + 2 ¶x ¶y ¶z Dr. Soha Albukhari 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 27 ¨ First postulate correlate between the state of the system and the wave function. ¨ Second postulate correlate between the property of the system and the quantum operator. Dr. Soha Albukhari Classical-Mechanical Observables and Their Corresponding Quantum-Mechanical Operators 28 Dr. Soha Albukhari 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 29 To obtain quantum operators: ¨ Write the classical expression of the property according to Newtonian Physics by knowing the Cartesian coordinated , momentum of motion & time ¨ Do the following modifications: 1. Time and Cartesian coordinates and each function depends on them leave unchanged. 2. Momentum of motion Pq is replaced by an operator d h −i! != ; dq 2π Dr. Soha Albukhari q : motion Cartitzian (x, y, z) 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 30 Quantum Mechanical Operator of the Kinetic Energy: ¨ the quantum operator of the kinetic energy (T) of a Particle of mass (m) move with a velocity (V) in the x-direction: 1. write the classical expression of the Tx: 1 T = mv 2 x 2 1. Since we are looking to insert the momentum motion, we will modify the above equation as follows: T = m 2 v 2 = p x2 x 2m 2m d 1. Replace each Px by the quantum operator - i! dx 1 æ d öæ d ö Tˆx = ç - i ! ÷ç - i ! ÷ 2m è dx øè dx ø 2 2 ! d Tˆx = - Dr. Soha Albukhari 2m dx 2 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 31 2 2 ! d Quantum operator of the kinetic Tˆx = - energy in the x-direction 2m dx 2 ¨ It does different from the Classical expression, since in the classical expression we could put the mass and motion values and thus obtain the numerical value of the kinetic energy. ¨ In the quantum expression, we can not put the values to obtain Tx, we have to do a mathematical operation (second derivative with respect to x) to obtain the numerical value of Tx. ¨ The same expression is obtained in the y-direction or z-direction. Dr. Soha Albukhari 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 32 2 2 ! d ! 2 d 2 Tˆy = - Tˆz = - 2m dy 2 2m dz 2 The Quantum Operator of the total kinetic energy is: ˆ ! 2 æ ¶ 2 ¶ 2 ¶ 2 ö Tˆ = Tˆx + Tˆy + Tˆz T =- çç 2 + 2 + 2 ÷÷ 2m è ¶x ¶y ¶z ø 2 ! Tˆ = - Ñ2 2m Where : æ ¶2 ¶2 ¶2 ö Ñ = çç 2 + 2 + 2 ÷÷ 2 è ¶x ¶y ¶z ø Dr. Soha Albukhari 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 33 Quantum Mechanical Operator of the Total Energy: ¨ The Classical expression of the total energy (E) is: E = T +V Hamilton Equation Where, T : Kinetic Energy , V : Potential Energy ¨ The Quantum Operator of the Total Energy Ĥ Hˆ = Tˆ + Vˆ ¨ The potential energy is a function in coordinates (position). So, the quantum operator of the potential energy is leaved without any change. ¨ Total Energy Quantum Operator is: !2 2 Ĥ = − ∇ + V̂ 2m Dr. Soha Albukhari 2.2 Second Postulate: Quantum Mechanical Operators and Observables Represent Classical Mechanical Variables 34 Classical Quantum Property Expression Expression Momentum motion in the d - i! x-direction mV dx 1 2 !2 2 Total Kinetic Energy mv - Ñ 2 2m Potential Energy V Vˆ 2 ! Total Energy E = T +V Hˆ = - Ñ 2 + Vˆ 2m Dr. Soha Albukhari 2.3 Third Postulate: Absolute Values of Sharp Observables (Observable quantities must be eigenvalues of quantum mechanical operators) 35 Sharp Property Is the property that its numerical value never change with changing of time or (Sharp Observable) position If the numerical value of this property is measured several times, the result will be the Absolute value same numerical value. (Eigen Value) This numerical value called Absolute value (Eigen Value) Third postulate How to calculate the Eigen value Dr. Soha Albukhari 2.3 Third Postulate: Absolute Values of Sharp Observables 36 Third postulate: In any measurement of the observable associated with the operator â , the only values that will ever be observed are the eigen values a, which satisfy the eigen value equation: aˆy = ay Eigen-value Equation α̂ : The quantum operator corresponding to the property being measured ψ : The wavefunction (Eigen function) a : The numerical value of the property (Eigen value) Dr. Soha Albukhari 2.3 Third Postulate: Absolute Values of Sharp Observables 37 aˆy = ay Eigen-value Equation ¨ Applying the mathematical operation of this operator on the state wavefunction results the state wavefunction without any changes times a constant. This constant is the desired numerical value. ¨ Generally, an operator will have a set of eigen functions and eigenvalues, and we indicate this by writing: α̂ ψ = a ψ n n n n ¨ Thus, in any experiment designed to measure the observable corresponding to â , the only values we find are a1, a2, a3,… The set of eigen values {an} of an operator â is called the spectrum of â Dr. Soha Albukhari 2.3 Third Postulate: Absolute Values of Sharp Observables 38 As a specific example, consider the measurement of the energy. The operator corresponding to the energy is the Hamiltonian operator, and its eigen value equation is: Hˆ y = Ey Schrödinger equation æ !2 2 ˆ ö The central equation çç - Ñ + V ÷÷y = Ey in quantum theory è 2m ø Determine the value of total energy: (Schrödinger equation) Dr. Soha Albukhari 2.3 Third Postulate: Absolute Values of Sharp Observables 39 Ĥ ψ = Eψ Schrödinger equation ¨ The solution of this equation gives the ψn and En. For the case of the particle in a box, En= n2h2/8ma2. Postulate 3 says that if we measure the energy of a particle in a box, we shall find one of these energies and no others. Notice that the Schrödinger equation is just one of many possible eigen value equations because there is one for each possible operator. Schrödinger equation is the most important and most famous one, however, because the energy spectrum of a system is one of its most important properties. Because of the equation ΔE = h ν, we see that the experimentally observed spectrum of the system is intimately related to the mathematical spectrum. This is the motivation for calling the set of eigen values of an operator its spectrum, and this is also why the Schrödinger equation is a special eigen value equation. Dr. Soha Albukhari 2.3 Third Postulate: Absolute Values of Sharp Observables 40 Example: Show that eikx is an eigen function of the momentum operator, What is the eigen value? Answer: d pˆ x = -i! ikx dx Apply the operator p̂ x on the function e d ikx p̂x e = −i! e = !keikx ikx dx p̂x eikx = !keikx eikx is an eigen function and the eigen value of the momentum operator is !k Dr. Soha Albukhari 2.4 Fourth Postulate: Expectation Values of Non-Sharp Observables 41 Non-sharp Property Is the property that its numerical value (Non-sharp change with changing of time or position Observable) If the numerical value of this property is measured several times, the result will be the Average value different numerical value. (Expectation value) This numerical value called Average value (Expectation Value) Fourth postulate How to calculate the Expectation value Dr. Soha Albukhari 2.4 Fourth Postulate: Expectation Values of Non-Sharp Observables 42 Fourth Postulate: If a system is in a state described by a normalized wave function ψ , then the average value of the observable corresponding to α̂ is given by a = ò yaˆydt òyy dt If the wave function is normalized: ò ydt = 1 y * ∗ a = ∫ ψ α̂ψ d τ Dr. Soha Albukhari 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation 43 ¨ The wave function in Postulate 1 contains time explicitly. the wave function is time-dependent. Y ( x, y , z , t ) ¨ The time dependence of wave functions is governed by the time-dependent Schrödinger equation. ¨ The derivation of time-dependent Schrödinger equation is difficult. ¨ We will take time-dependent Schrödinger equation as its, and show it is consistent with the time-independent Schrödinger equation Dr. Soha Albukhari 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation 44 Postulate 5: The wave function or state function of a system evolves in time according to the time-dependent Schrödinger equation. ˆ ¶Y ( x, t ) HY ( x, t ) = i! ¶t Time-Dependent Schrödinger Equation Dr. Soha Albukhari 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation 45 ˆ ¶Y ( x, t ) The time-dependent Schrödinger equation HY ( x, t ) = i! ¶t "2 2 Hamiltonian operator ∵ Ĥ = − ∇ + V̂ 2m ¶Y( x, t ) !2 2 i! =- Ñ Y( x, t ) + V ( x, t )Y( x, t ) ¶t 2m \V ( x, t ) = V ( x) Consider time-independent potential energy ¶Y( x, t ) !2 2 i! =- Ñ Y( x, t ) + V ( x)Y( x, t ) ¶t 2m Dr. Soha Albukhari 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation 46 Apply the Method of Separation of variables ψ − function − in − x Ψ(x, t) = ψ (x) f (t) f − function − in− t Substitute this equation into time-dependent Schrödinger equation ¶y ( x) f (t ) !2 2 i! =- Ñ y ( x) f (t ) + V ( x)y ( x) f (t ) ¶t 2m Rearrangement ¶f (t ) !2 2 i!y ( x) =- Ñ y ( x) f (t ) + V ( x)y ( x) f (t ) ¶t 2m Dr. Soha Albukhari 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation 47 ∂2 ∂2 ∂2 ∵∇ = 2 + 2 + 2 2 ∂x ∂y ∂z ¶f (t ) !2 i!y ( x) =- f (t )Ñ 2y ( x) + f (t )V ( x)y ( x) ¶t 2m Divide both sides by: y ( x) f (t ) 1 ¶f (t ) !2 1 1 i! =- Ñ y ( x) + 2 V ( x)y ( x) f (t ) ¶t 2m y ( x) y ( x) Dr. Soha Albukhari 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation 48 1 ¶f (t ) 1 é !2 2 ù i! = ê- Ñ y ( x) + V ( x)y ( x)ú f (t ) ¶t y ( x ) ë 2m û The left side is a The right side is a function of t only function of x only ¨ Both sides of the equation equal constant value (E) postulate 3! ¨ Solve each side separately Dr. Soha Albukhari 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation 49 The right side of the equation: 1 é !2 2 ù ê- Ñ y ( x) + V ( x)y ( x)ú = E y ( x ) ë 2m û é !2 2 ù ê - Ñ y ( x ) + V ( x )y ( x ) ú = Ey ( x) ë 2m û é !2 2 ù ê - Ñ + V ( x ) úy ( x) = Ey ( x) ë 2m û !2 2 - Ñ + V ( x) = Tˆ + Vˆ = Hˆ 2m Time-Independent \ Hˆ y ( x) = Ey ( x) Schrödinger Equation Dr. Soha Albukhari 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation 50 Schrödinger Equation ˆ ¶Y ( x, t ) Time-Dependent HY ( x, t ) = i! Schrödinger Equation ¶t Time-Independent Ĥ ψ (x) = Eψ (x) Schrödinger Equation Dr. Soha Albukhari 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation 51 The left side of the equation: 1 ¶f (t ) i! =E f (t ) ¶t Rearrange and multiply right ∂f (t) = − i Ef (t) side of the equation by i/i ∂t ! By integration: f (t ) = e -iEt / ! And so Ψ(x,t) is of the form: Ψ(x, t) = ψ (x)e−iEt/! ψ (x) Solve this function e−iEt/! Multiply the result by this function Dr. Soha Albukhari 2.5 Fifth Postulate: The Time-Independent Schrödinger Equation from the Time-Dependent Schrödinger equation 52 There is a set of solutions from last equation: Ψ n (x, t) = ψ n (x)e−iEnt/! n = 1, 2, 3,… The probability density and the averages calculated from this equation are Independent of Time. Stationary state wave functions y n (x) Stationary State: If the properties of the system in one of its state do not change with time, the system is said to be in a stationary state. Stationary State Wave Function: is the wave function of a stationary state. Dr. Soha Albukhari 53 ∞ ∫ −∞ dτΨ*(r, t)Ψ(r, t) = 1 , dτ = dxdydz Normalized wave function a = ∫ ψ ∗α̂ψ d τ ¶Y ( x, t ) Hˆ Y ( x, t ) = i! ¶t Dr. Soha Albukhari Chapter 2 Assignment 54 Write your own two essay questions. You need to write the questions and answers. Dr. Soha Albukhari

CHEM 242 Chapter 2.pdf

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