Time dependent Schrodinger equation

Lesson: Time dependent Schrodinger equation Lesson Developer: Dr. Monika Goyal, College/Department: Shyam Lal College (Day), University of Delhi

Table of contents 1.1 Introduction 1. Dynamical evolution of a quantum state 1.3 Concept of Stationary States 1.4 Phase and Group velocities and concept of wave packets 1.5 Properties of a wave function 1.6 Operators and Expectation values

Current unit should make you understand the following 1. What is a free particle wave function?. What is Time-evolution in quantum mechanics? 3. What are wave packets and dynamical evolution of a wave packet 4. Various properties of a well-behaved wave function 5. What is an average or expectation value of an observable? 6. Operators associated with observables. 1.1 Introduction All physical systems evolve in time. The change in the dynamical properties of any quantum mechanical system is termed as the time evolution of a quantum state. For a quantum state being described by a wave function ψ(x, 0), at t=0, {that is, a quantum particle which is moving in x-direction and is represented by a wave at some initial time t=0}, a physical law that tells us what the state will be at some other time, t, is a quantum law of evolution known as time-dependent Schrodinger equation. This equation tells us how the initial information about the system changes with time according to a particular physical circumstance that a system finds itself in. The solution of this equation gives us the wave function as the time passes in a particular physical situation. The information about the position, momentum and energy of a particle at any time can then be obtained from the knowledge of this wave function which we will understand as we slowly develop the theory of quantum mechanics.

ERWIN SCHRODINGER [a] (1887-1961) was born in Vienna to an Austrian father and a half-english mother. He was taught at home by private teachers until he was 11 years old and then attended Vienna s Akademisches Gymnasium. He went on to enter the University of Vienna where at he received his doctorate. After World War 1, during which he served as an artillery officer, Schrodinger got appointments at several German universities before becoming professor of physics in Zurich, Switzerland. In an epochal paper on Quantization as an Eigenvalue Problem, in 196, Schrodinger introduced the equation that bears his name and solved it for the hydrogen atom, thereby opening doors to the modern view of atomic physics. Within months, he applied wave mechanics to the harmonic oscillator, the diatomic molecule, the hydrogen atom in an electric field, the absorption and emission of radiation and the scattering of radiation by atoms and molecules. In 197, he succeeded Planck at the University of Berlin but left Germany in 1933, the year he received the Nobel Prize, when Nazis came to power. He was at Dublin s Institute for Advanced study from 1939 till 1956. In Dublin, he became interested in biology, in particular, the mechanism of hereditary. Schrodinger s 1944 book What is life introduced biologists to a new way of thinking-that of a physicist. It was his attempt to link quantum physics and genetics. [a] Concepts of Modern Physics, 6 th edition, Arthur Beiser, Tata McGraw Hill Edition 1. Dynamical evolution of a quantum state Since any quantum particle can be represented as a wave (from our knowledge of wave-particle duality and de-broglie relation 1 ), we start with considering a free quantum particle moving in one dimension. For simplicity, let us take x direction as the direction of motion and see how its wave function is developed with time.

From our knowledge of waves, we can write a standard wave function for a particle moving in +x direction as ( x, t) Ae i( kxt ) (1) Here A is an arbitrary constant. Since we already know the following relations 3 v,, k h E h, p p k; E () Again writing the wave function in another form using () ( i )( Et px) ( x, t) Ae (3) Here the symbols have their usual meanings. Let us differentiate (3) twice with respect to x p x p x x (4) Again differentiate (3) w.r.t t ie t Et i t (5) Since the total energy ( E ) of a non-relativistic free particle is given by p E V ( x, t) (6) m Multiplying (6) by Ψ(x,t) and substitute from (4) and (5) we get

( xt, ). ( x, t) V ( x, t) ( x, t) i m t (7) The above equation is famous time dependent Schrodinger equation. In the development so far, we see that Schrodinger equation is a linear differential equation in the wave function. It is an extremely useful property since if the equation is linear it allows us to add simple contributions from individual elements to get a solution for a more complex circumstance. The solution of the Schrodinger equation for a particle in a given physical circumstance gives a wave function which contains all the information about the particle with time. 1.3 Solution of a Schrodinger equation or Stationary states When the potential in a quantum mechanical problem is time-independent (for e.g. linear harmonic oscillator-an application of Schrodinger equation * ), the general solution of the Schrodinger equation can be expressed as a product of a function of space coordinates and a function of time coordinate and the solution thus obtained is known as separable solution. Let us write ( x, t) ( x) ( t) (1) If we substitute (1) in equation (7) of previous section and then divide the whole equation by ( x) ( t), we get 1 d ( x ) 1 ( ) ( ) d V x i t m( x) dx ( t) dt () In this equation, the left hand side is a function of space coordinate only and the right hand side is a function of time coordinate only, hence both sides must then be equal to a same constant. Focusing on the right hand side, we see if i 1 d() t B () t dt (3) 1 Topics covered in Modern Physics Wave equation from our knowledge of waves and oscillations 3 Meaning of symbols is depicted at the end of the lesson *Applications of S.E will be discussed in subsequent units

Where B is a constant which needs to be determined, then we can solve the above differential equation to get ibt ( t) (0) exp (4) Here, ɸ(0) is the value of the time function at t=0 or we say it is some constant initial value which can be absorbed in ψ(x). The value of the constant B, can be determined from the operator representation of physical variable E i-e total energy of the particle. (Operators will be discussed in the last section of this lesson). For the time being, let us write the operator form of ENERGY (E) d x, t i E x, t or dt t d i E t dt (5) Comparing equation (3) and (5), we get B=E. Hence the separable solution to the time-dependent Schrodinger equation is written as iet ( x, t) ( x)exp (6) Since the Schrodinger equation is linear in wave function, any linear combination of solutions will also be a solution as suggested by SUPERPOSITION PRINCIPLE. This principle will be discussed in detail as we move on with our theoretical buildup of quantum mechanics. A free particle can have any position, therefore, for a particle that can be in any one of the infinitely many discrete positions; the superposition principle tells us the most general solution to the time-dependent Schrodinger equation as n1 n ( x, t) a ( x) e n n ie t / (7) where, the coefficients a n are arbitrary complex constants. In the above equation, the summation runs over discrete states. The system described by a wave function of this form is in a STATIONARY STATE, since the probability density Ψ(x,t) is independent of time. The probability density in one dimension is defined as

* ( x, t) ( x, t) ( x, t) dx For a stationary state, the probability density will become * iet iet * ( x, t) ( x) e ( x) e dx ( x) dx Which is only space dependent and time dependence cancel out. All observables which are time independent like x, p, E etc. have time-independent EXPECTATION VALUES in the stationary states. Expectation values will be discussed in section 1.6. Application of S.E to a particle confined in a 1-dim box of width L Here boundary conditions being ( x 0, t) ( x L, t) 0 And V at x=0, L and V=0 inside the box. For time independent potential, the solution is given by equation (6) wherein ψ(x) satisfies timeindependent Schrodinger equation, hence we can write for this particular example our eq. as d ( x) m E x dx ( ) 0 (1) Writing general solution of (1) 1 1 me me ( x) Asin x B cos x () Using boundary conditions, we get B=0 and 1 me L n (3) Or n E ml (4)

And ( ) sin n x x L L (5) Value Addition Example 1: Expectation value of momentum of a stationary state is ZERO. Given a stationary state for a particle in an infinitely deep potential well of width L d V ( x) ( x) E ( x) m dx p dx i dx * d n x L L ( ) sin * ( ) n x x (See section 1.6) (Normalized wave function of a particle in a box) d n nx sin dx L L L p L * d 0 i dx dx L n sin n x cos n x dx i L L L L 0 p L n x sin 0 il L 0 1.4 Phase and Group velocities For a free particle, V(x) = 0, that is, no forces are acting on the particle and thus from timedependent Schrodinger equation and operator representation of E which is equation (5) of the previous section, we can write

m d ( x, t) E ( x, t) (1) dx Or d ( x, t) dx k ( x, t) where me k () Equation () is a second order partial differential equation whose general solution can be written as ikxiet / ikxiet / ( x, t) Ce De (3) Here C and D are arbitrary constants which can be determined from boundary conditions. Since there are no boundary conditions to restrict the possible values of E; hence a free particle can carry any energy and the time dependence is written as exp( iet / ). This is referred as a quantum state for a free particle. Substituting the value of E from () kt kt ik ( x ) ik ( x ) m m ( x, t) Ae Be (4) For a particle moving in +x direction kt i( kx ) m ( x, t) Ae (5) Where k and m v quan k k m v quan k E m m ; (6) Now, the classical speed of a free particle with energy E determined from the relation of kinetic energy is given by p m v E E ; v (7) m m m Here from (6) and (7), we see that the velocity of the quantum mechanical wave function is half the velocity of a free particle! Hence it is a PARADOX and we are forced to ask ourselves if separable solutions actually represent physically realizable states for a free particle? This paradox was answered by the concept of WAVE PACKET. Now what is a WAVE PACKET?

To understand WAVE PACKET, let us first understand how we obtain the Fourier transform of a wave function. For simplicity, we again restrict ourselves to one dimension. Let us consider a periodic square wave with a repeat distance λ (Fig.1). Any such function can be expressed as a Fourier series 4 nx f ( x) Fn cos( ) or (8) n0 f ( x) Fncos( knx) n0 (9) Where F n are Fourier constants. As we can see from expression (8), the various terms in the Fourier series corresponds to different wavelengths; for n=1, we have a term with longest wavelength known as fundamental term and as n take higher values the subsequent terms are called higher harmonics. The corresponding wavenumbers will then be * 3* 4* k1, k, k3, k 4 And so on such that k = π/λ. From our knowledge of de-broglie relation which relates h wavelength with momentum, that is,, we can say, if we are talking of k(λ)-space, we are p in a way talking of momentum space and Fourier transform of a wave function is actually a kind of its representation in momentum space. Now let us consider a non-periodic function. A non-periodic function means a function which does not repeat itself after regular intervals so it can be considered a periodic function in the limit that it repeats itself after an infinite interval. Or in other words, for a non-periodic function, since the interval λ goes to infinity, the distance between k s in the Fourier sum becomes infinitesimally small approaching zero (Fig.1). Fig.1

4 Topic explained in a lesson on mathematical physics Thus, a summation over k in case of periodic function converts into an integral over k for a nonperiodic function, hence f ( x) F( k)cos kxdk Or f ( x) F( k) e ikx dk (10) The function F(k) is called the Fourier transform of f(x). Now, let us come back to the general solution to the time-dependent Schrodinger equation which is nothing but a linear combination of separable solutions and is written as n1 n ( x, t) a ( x) e n n ie t/ (11) For continuous distribution, the sum would be replaced by an integral in k-space k i( kx t) m 1 ( x, t) ( k) e dk (1) where ɸ(k) is a Fourier transform of the wave function Ψ(x,0). 1 ikx ( x,0) ( k) e dk (13) 1 ikx And ( k) ( x,0) e dx (14) Equation (1) is an integral over the continuous variable, k, instead of a sum over, n, as in 1 equation (11) and plays the role of a n in equation (11) which is a NORMALIZATION CONSTANT (discussed in section 1.5). This wave function carries a range of, k s, and hence a range of energies. This is known as WAVE PACKET. Hence a wave packet is defined as a sinusoidal function whose amplitude is modulated by ɸ(k).

Fig. It consists of ripples contained within an envelope. So the quantum mechanical representation of a moving particle is not a single wave but a wave packet. As fig. depicts, the individual ripples inside an envelope move with the velocity called PHASE VELOCITY while the envelope move with the velocity known as GROUP VELOCITY. For a free particle wave function, we will see that the group velocity is twice that of the phase velocity and the classical particle velocity matches up with the group velocity of the wave packet. Now, let us return to the PARADOX which says that separable solutions travel at a wrong speed for the particle it represents which was answered by the concept of wave packets. The group velocity of a wave packet with a general form 1 i( kxt ) x t k e dk (, ) ( ) (15) is defined as vg d and phase velocity as vp dk ; k k Since, m d k k and (16) dk m k m Hence we conclude that v v g p which answers the PARADOX. Any wave packet can be formed by superposition of many different sinusoidal waves or a wave function can be written as a sum of exponentials which contain sine or cosine. A wave packet is a non-periodic function which means a function which does not repeat itself after regular intervals. But as stated above, any non-periodic function can be treated as periodic when we take the limit that repeat distance is infinite. Equation (1) is a typical expression which represents a wave packet. As an example of timeevolution of quantum state, let us examine Gaussian wave packet which is described in detail in

our unit on time-independent Schrodinger equation. For the time-being let us focus on its timeevolution or how does it evolve with time. Example : Dynamical evolution of a Gaussian wave packet At time t=0, any free particle is described by a Gaussian wave packet, which is 1 i ( x,0) e exp( p0x) ( ) x 0 1 4 0 (17) The wave function is normalized ( x,0) dx 1 x 0 1 0 x x x p p 0 (18) (19) (0) (1) p p p 0 () Equation (17) describes a particle which is localized within a distance ~ σ 0 around x=0 and moving with an average momentum of p 0 (in +x direction) with a momentum spread 0. Equation (18) represents the normalization condition for a wave function, equation (19) is the average or the expectation value of position x, equation (0) is the uncertainty in the position x, equation (1) is the expectation value of momentum p and equation () is the uncertainty in momentum p. All these will be discussed in detail in our unit on time-independent Schrodinger equation while Operators and expectation values are discussed in the section 1.6. Calculating ɸ (p) using equation (14)

Time dependent Schrodinger equation x 1 1 i 0 ( p) e exp ( p p0) x dx 1 0 1 4 0 ( p p0 ) 0 exp Using the identity (3) e dx exp 4 x x (4) Substitute for ɸ(p) in Ψ(x,t) to get 1 ( X ) i ( X, ) exp 1 1 X ( X ) 4 (1 ) (1 ) 0(1 i ) where x t p0 0 X,, m 0 0 (5) m is the mass of the particle. This represents the dynamical evolution of a Gaussian wave packet. 1.5 PROPERTIES OF A WAVE FUNCTION A quantum particle can be represented as a wave and the amplitude of the wave is known as its wave function which is generally denoted by a symbol, Ψ(r,t), where r is the position of the particle in 3 dimensions. If the wave function is zero at a point then we say the particle cannot be found at that point. Although, there is no way we can measure the amplitude of a wave function of a particle but the quantity that can be measured is its PROBABILITY DENSITY which is defined as the probability of finding the particle in a small volume dτ at a time t, and is represented as * P ( r, t) d ( r, t) ( r, t) d (1)

Here Ψ * is the complex conjugate of the wave function. This is also called fundamental postulate of quantum mechanics. The physical significance of a wave function is its probability density which is a measurable quantity. Example 3: The wave function of a particle is n x sin. What is the probability of finding the particle between 0.50L and 0.55L L L for n=1. Solution: From the definition of probability density, we know x x n ( ) sin L x1 x1 n x P x dx dx L Here, x 1 =0.50L and x = 0.55L and for n=1, we have P = 9.9 percent Other conditions for physical acceptability of a wave function are as under: (1) A well behaved wave function always obeys the Schrodinger equation. () It must be finite, single-valued and continuous everywhere. (3) The first three partial derivatives of the wave function must also be finite, single-valued and continuous everywhere. (4) The probability of finding the particle somewhere in the region - to + (in 1 dimension) is given by * ( x, t) dx ( x, t) ( x, t) dx 1 () This is referred to as the NORMALIZATION CONDITION. Any well behaved wave function must be normalized. (5) Any physical quantity which can be experimentally measured is referred to as an OBSERVABLE. Each observable is associated with an OPERATOR. Let O be any operator associated with an observable. Then, the AVERAGE or EXPECTATION VALUE of O is given by O ( x, t) O( x, t) dx (3)

Where Ψ(x,t) is a normalized wave function for a particle restricted to only x direction. For example: The average or expectation value of momentum of a particle is given by p ( x, t) pˆ ( x, t) dx (4) Where ˆp is momentum operator discussed in detail in section 1.6. (6) Eigenvalue equation H Ψ(r,t) = E Ψ(r,t) (5) Here H is the Hamiltonian (total energy) operator and E is the corresponding energy value or Eigen value (see section 1.6). If more than one state or wave functions which are linearly independent corresponds to the same energy E, then the states are said to be degenerate. Any linear combination of these states a a a 1 1 g g is also an Eigen function with the same energy eigenvalue. Also all energy eigenvalues E n are real and if E E n m The corresponding Eigen functions are necessarily Orthogonal, that is m ndx mn where δ mn = 0 for m n =1 for m = n (6) 1.6 OPERATORS AND EXPECTATION VALUES A free particle wave function in x direction is given by ( i )( Et px) ( x, t) Ae (1) Differentiate equation (1) with respect to x and t ( x, t ) i p ( x, t ) x p( x, t) i ( xt, ) x ()

( x, t) i E ( x, t ) t Time dependent Schrodinger equation E( x, t) i ( xt, ) t (3) On carefully examining equation () and (3) we see that, operators are associated with dynamical quantities p and E. Operator corresponding to observable p is i x Operator corresponding to observable E is i t Hence, in general, we write an operator corresponding to any observable (a physical quantity which can be measured) with a cap sign on its symbol. Now, let us write the total energy operator The total energy of a particle is the sum of its kinetic energy and potential energy E K. E V Writing in operator form Eˆ K. Eˆ Vˆ (4) ˆ pˆ 1 KE. m m ix m x and (5) Ê i t (6) Substituting equation (5) and equation (6) in equation (4) and multiplying by Ψ We get i V t m x (7) This is nothing but time-dependent Schrodinger equation. Rewriting the above equation V i m x t

The term in the bracket on L.H.S is the total energy operator or the Hamiltonian operator V Hˆ m x (8) So the time-dependent Schrodinger equation in operator form is written as H E (9) This is known as EIGEN VALUE equation for total energy. More generally, any random dynamic variable S (say), may attain quantized values S n (say), also known as Eigen values corresponding to that observable, if the wave function Ψ n of the system be such that it satisfies Sˆ S (10) n n n Here, each S n is a real number. We specify an operator by specifying its operation on any wave-function ( xt, ) Value Addition Example 4: An eigenfunction of the operator d /dx corresponding eigenvalue. is 3x e. Find the Sol. Here, d 3x d d 3x d 3x 3x Sˆ ( e ) ( e ) (3 e ) 9e dx dx dx dx But e 3x S ˆ 9, so Hence, the eigen value S here is 9. pˆ x i p x Where p represents the Eigen value (11)

Now, let us talk about EXPECTATION VALUES. Since we know that Ψ * Ψ is associated with position probability density, the expectation value or average value of the position of a particle or its x coordinate is given by x xd d (1) where integration runs over the entire volume. If the particle is restricted to a single dimension then accordingly the integral into consideration becomes a line integral. For a normalized wave function, the denominator in equation (1) becomes unity and hence the average or expectation value of x is then given by (13) * x xd Finding expectation value of E, that is, E p V (14) m Let us go back to the Schrodinger equation, premultiply it by Ψ * and then integrate over entire volume i d (, ) t m d V r t d (15) Equation (14) and (15) are same if E i d t (16) Hence, in general for finding the expectation value of any dynamical quantity or observable, we operate its operator representation on the wave function, Ψ, premultiply by Ψ * and then integrate over the volume considered. Thus px i d x (17)

p d x (18) x Example 5: Let us take two operators ˆx and p ˆ x and define the operator xp ˆ x such that xpˆ x xi i x x x (1) Now let us take ( x) p ˆ xx i i i x x x () We notice that xpˆ x and px ˆ x are not equal or the order in which the operators are multiplied is important. We call it as multiplication of ˆx and pˆ x is non-commutative. OR The commutator of position and momentum operators is xpˆ pxˆ i x, p i y, p z, p x y z (3) Example 6: Find out the normalized Eigen functions of the Operator pˆ x i d dx ; Also ip exp x x ' dp ( x x '), prove that (1) Using 3 ip. r ip '. r 3 1 e e d r ( p p ') () Solution: From Momentum Operator representation, we know

d i p dx The solution of the above differential equation yield momentum space wave functions. ipx p( x ) Ae Where A is a constant which can be determined from normalization condition and it gives A 1 Hence 1 ( x p ) e ipx/ In 3 dimensions 3 ip. r / 1 ( r p ) e and 3 1 ip'. r / p' ( r) e Now, 3 * 3 1 ip'. r / ip. r / 3 p' ( ) p( ) r r d r e e d r From condition (1) given in the question, equation (3) can be written as equal to 3 ip. r ip '. r 3 1 e e d r ( p p ')

Value addition Here δ(x-x ) is Dirac delta function which is defined as equal to zero if x is not equal to x and is infinite at x=x such that ( x x') dx 1 It is an infinitely high, infinitely narrow spike of area 1 at the point x=x. if we multiply δ(x-x ) by an ordinary function f(x), it is same as multiplying by f(x ). That is, f ( x) ( x x') dx f ( x') ( x x') dx f ( x') Domain of integration need not be from - to, but it should be such that it includes point x. UNSOLVED PROBLEMS 1. The wave function of a particle in a stationary state with an energy, E, at time t=0 is ψ(x). After how much time will the wave function again be ψ(x)?. Write time-dependent Schrodinger equation in 3 dimensions. 3. If a particle of mass m moving in a one dimensional potential V(x), satisfies V(x)=V(-x) and ψ(x) is a stationary state with energy E, then show that Ψ (- x) is also a stationary state with the same energy. 4. Plot the probability density as a function of x for a particle confined in a box for (i) ground state (ii) first excited state (iii) second excited state.

ˆp ˆ pˆ K 5. What are the eigenvalues of the operators and m, operating on momentum space wave functions? What are the completeness and orthonormality conditions for 6. momentum space wave functions? 7. Using the operator representations of px, py, p z, prove that x, py 0 y, p p, p 0 p, p x y z x z 8. What are the Eigen functions and eigenvalues of the operator ˆx? 9. What is the distance between the closest energy levels for a particle in infinite deep potential well? 4 Nxe x 10. Find normalization constant N for the wave function. 11. Attempt the following MCQ s For a free particle, the wave function ψ is a a) Quantity which is real b) quantity which is complex c) quantity which is imaginary d) Any one of these 1. For stationary states a) Probability density is time dependent b) The average values of time-independent operators are time dependent c) The most general solution is a superposition of separable solutions d) All of the above 13. The dynamical evolution of the expectation values of position, momentum and energy of a quantum mechanical particle is given by a) Maxwell s Equation b) Green s Theorem c) Gauss - Divergence Theorem d) Schrödinger Equation 14. Group velocity is defined as a) Velocity of a single wave with definite wavelength

b) Velocity of a group formed by many waves c) Velocity of a free particle d) None of these 15. Time dependent Schrodinger equation a) is one of the postulate of quantum mechanics b) Equates second order space derivative with first order time derivative c) Is the Eigen Value equation for the energy operator (Hamiltonian Operator) d) all of the above LIST OF SYMBOLS USED IN THE LESSON Ψ = wave function, t = time, λ = wavelength, = frequency, k = wavenumber, = angular frequency, E = energy, h = Planck s constant, x = position of the particle in x-direction, p = momentum, V = Potential, m = mass, dimensions, = volume element Suggested Readings x y x, r = position vector in 3-1. Introduction to Quantum Mechanics, D.J. Griffith, nd Ed. 005, Pearson Education. Quantum Mechanics: Theory and Application, A. Ghatak and S. Lokanathan, Springer Science 3. Quantum Mechanics, Eugen Merzbacher, 3 rd Ed. 004, John Wiley & Sons, INC. 4. Quantum Mechanics for Scientists and Engineers, David A.B. Miller,008, Cambridge University Press. 5. Quantum Mechanics, Leonard I. Schiff, 3 rd Edn 010, Tata McGraw Hill. 6. Quantum Mechanics, Robert Eisberg and Robert Resnick, nd Edn,00,Wiley.