Introduction to Quantum Mechanics I
Introduction to Quantum Mechanics I PHY 567
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This 4 page Class Notes was uploaded by Ms. Bryce Wisoky on Wednesday October 21, 2015. The Class Notes belongs to PHY 567 at Syracuse University taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/225636/phy-567-syracuse-university in Physics 2 at Syracuse University.
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Date Created: 10/21/15
Lecture 4 Ehrenfest s theorem The connection to classical physics can be made more explicit in a celebrated theorem due to Paul Ehrenfest which shows that quantum expectation values evolve according to Newton s 2nd law Consider the time rate of change of the expectation value of the momentum lt p gt dltPgt 2 2 dt dl8tqji8xqj Using Leibnitz we nd dltpgt dx8j i It at 81 8x87 Substituting in from the Schroedinger equation and using integration by parts we can SOC d ltp gt dt The right hand side is nothing but the expectation value of the force on the particle and we have the result Similarly it is to see that dltxgtlt gt dt p Summary so far QM tells us that the most information we can obtain about a microscopic particle is contained in its wavefunction x Once we know the wave function at one time Schroedinger s equation allows you to calculate it at any later time Physical particles are described by normalized wavefunctions flIll 1 In fact QM tells us to interpret llll a probability density This then allows us to write expressions for the expectation value of some observable quantity Q l p lt Q gt Vow where the QM operator Q is just obtained from its classical expression by replacing x with just 1 and p by Time independent Schroedinger equation OK we have the Schroedinger equation but how do we go about solving it It turns out that if the potential V is independent of time this may be accomplished by a method termed the separation of variables What this means is that we seek solutions of the form We 75 Motif The justi cation for this is three fold o More general solutions can be built up from these separable solutions 0 They turn out to be states of de nite energy 0 Expectation values in these states are independent of time They are also termed stationary states If we do this we nd that the Schroedinger equation reduces to two ordinary differential equations ft exp iEth 1 MW E l 2 E is a constant which we will identify shortly the energy of the state and H is the Hamiltonian 72 d2 I a H 2m l This second equation involving the Hamiltonian 2 is called the timeindependent Schroedinger equation Notice that advertised the probability density V is time independent Furthermore the operator H is a QM version of the classical energy function for the system and from eqn2 has an expecta tion value equal to E Furthermore any power of H has expectation value just equal to E raised to that power Thus the variance of the probability distribution for the energy is zero the distribution is trivial Thus any mea surement of the energy will return exactly E stationary states are states of xed energy But what is the energy E so far we have not speci ed it In general we will see that the energy E can take on an in nite number of discrete values dependent on the nature of the potential V We will call these values E1 E2 E3 and to each allowed value of E there will be an associated solution to the timeindependent Schroedinger equation p1xp2x It is a theorem we will not attempt to prove it that the most general solution to Schroedinger s equation is a linear combination of these stationary state solutions the cn s are constants x t Z cn nl E iEquot h These coefficients on can be usually found from a knowledge of the wave function at t 0 and the solution of the timeindependent problem The moral of the story is that once we have solved the timeindependent equa tion we have very little left to do to find the most general solution to the time dependent Schroedinger equation Furthermore the timeindependent equation does not contain i and so we can just look for real solutions of this equation Examples The in nite square well Suppose V 0 for 0 g x g a and is in nite elsewhere A particle is permanently confined inside this potential well It could be thought of a very crude model for a single electron atom Classically a particle confined to such a system would just bounce back and forth at constant speed Its energy could take on any value We will see that in QM the allowed possible energies are discrete First notice that p 0 for 1 lt 0 and 1 gt a since there is no probability of finding the particle outside the well Inside the well where V 0 the timeindependent equation reduces to 4 d2 2m 112 E Assuming that E gt 0 we may introduce the variable k xQmEh2 and write this equation 2 L k2ltp dxz The general solution to this is Asinkx8coskx 3 The constants A and B are xed by applying the boundary conditions M0 0 and Ma 0 This yields 8 0 and the quantizatian canditian sin ka 0 The latter means that ka mr Thus not all wavelengths are allowed only those which correspond to standing waves in the well We have seen that the Bohr quantization condition for the hydrogen atom could be understood on a similar basis here we see for the rst time that the formal theory of QM is able to explain many of the quantum phenomena which had been observed and which had proven so difficult for classical physics to account for What remains well we still have to normalize the solution that is the origin of the remaining freedom in the constant A Thus we nd that the stationary states of this potential are of the form sinnlx an a a The energy of this state is En We speak of the ground state39s the state of lowest energy which here corresponds to n 1 with E1 Classically the state of lowest energy corresponds to the particle at rest with E 0 We see in QM that such a state is impossible The minimum energy the particle can have is El which increases we con ne the particle to smaller and smaller regions a gt 0 This is a corollary of a very general theorem in QM called Heisenberg s uncertainty principle This roughly states that the more accurately ones knows the position of a quantum particle the less certain we are about its momentum and hence its energy In fact the product of the uncertainty in its position times the uncertainty in its momentum is always greater than some minimum which is equal to hQ We will prove this theorem later in the course but one immediate corollary is that no particle can ever be at rest at a point since then it would have a wellde ned position and momentum zerol Thus even at zero temperature particles always su 39er fluctuations in their positions and momenta they are somewhat smeared out This of course is required if they sometimes behave like waves Other points to notice 91 increases the number of zero crossings of the wavefunction increases By symmetry the expectation value for x is at x aQ for all states
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