sign in sign up
<<
Home , Uncertainty principle

Class 18: The Schrödinger equation

In , the dynamics of a single of mass m acted on by a conservative force is determined by the Schrödinger equation

ℏ2 ∂ψ −∇+2ψV()r ψ = i ℏ . (18.1) 2m∂ t

Here V(r) is the potential associated with the force and ℏ is the reduced Planck’s constant. The probability density, P(r), that the particle is located at r is determined from the function, ψ (r), by

P(r) =ψ* ψ , (18.2) where ψ * is the of ψ. To get some idea of the significance of the Schrödinger equation and the , consider a for which V(r) = 0. Now the Schrödinger equation has the form of a diffusion equation

iℏ ∂ψ ∇2ψ = . (18.3) 2m∂ t

with diffusion coefficient iℏ / 2 m . Hence ℏ m is a measure of the rate of spreading of the wave

function. Classically a particle at rest at a definite position r0 remains at rest at the same position. This corresponds to the limit ℏ = 0. The wave function in this case is zero everywhere except at

r0. If ℏ ≠ 0, then the ‘’ in position grows with time. This spreading is illustrated below for a moving particle.

1.0 Equal time intervals

0.8

0.6

ψ ∗

ψ 0.4

0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0 x Here the Schrödinger equation has been solved to find how the wave function changes with time. 1/2 We see that as time progresses the width of P(x) increases ∝ ()ℏt/ m . This width is proportional to the of the position x. In the standard deviation of the of a variable is often called the uncertainty in the variable. The reduced Planck’s constant ℏ has the value 1.05 10 -34 J s. If we have an object of mass 1 kg, then the time scale for the uncertainty in the position of this object to increase by about 10 -6 m is estimated to be 3 10 6 years. Hence classical is amply adequate to describe the dynamics of macroscopic objects. Wave packet spreading can also be understood by making use of the Heisenberg Uncertainty . Suppose at time t, the uncertainty in the position of a particle of mass m is ∆x. The uncertainty in the velocity of the particle is related to the uncertainty in the particle’s by ∆v = ∆ p m . Hence the uncertainty in position increases with time at rate

d∆ x ∆ p ℏ = = , dt m2 m∆ x where in the last step the uncertainty in the momentum has been taken to the minimum allowed by the Heisenberg Uncertainty Principle. If the uncertainty at time zero is ∆x0 , then

ℏt ∆x2 =∆ x 2 + , 0 m which is consistent with the result shown above. We conclude that the Heisenberg Uncertainty Principle is embedded in the Schrödinger equation. On the atomic scale, classical physics fails. The , Compton scattering, and double slit experiments show that have both wave-like and particle-like properties. Similarly, the Davisson – Germer experiment shows that also have both wave-like and particle like properties. This led to the idea of wave-particle duality . Different experiments elicit different behavior but both a wave-based picture and a particle-based picture are necessary and complementary. The melding of these two pictures is achieved by quantum mechanics. A ‘particle’ can be described as a wave with all the dynamic information of the particle being contained in the wave function ψ. Consider a one-dimensional monochromatic plane wave

ψ=expi( kx − ω t )  , (18.4)

(we assume that all can be built-up from such plane waves). Here k is the wave number and ω is the angular frequency of the wave. From the de Broglie relations, the of a particle with frequency ω is

E = ℏω. (18.5)

To relate this to the wave function, we note that

∂ψ =−iωexp i() kx − ω t  =− i ωψ . (18.6) ∂t Introducing the energy, we find ∂ψ Eψ = i ℏ . (18.7) ∂t Hence the energy is associated with the ∂ iℏ . (18.8) ∂t Also from the de Broglie relations, the momentum is related to the wave number by p= ℏ k . (18.9) Since ∂ψ = ik ψ , (18.10) ∂x we find ∂ψ pψ = − i ℏ , (18.11) ∂x so that the momentum is associated with the operator ∂ −iℏ . (18.12) ∂x For a free particle the energy is purely kinetic and is related to the momentum of the particle by p2 E = . (18.13) 2m If we consider this equation in terms of the operators and apply it to the wave function, we find ∂ψℏ2 ∂ 2 ψ iℏ = − . (18.14) ∂t2 m ∂ x 2 which is the Schrödinger equation for a free particle. If we include a potential energy in the total energy, p2 E= + V() x , (18.15) 2m we have ∂ψℏ2 ∂ 2 ψ iℏ = − + V() x ψ . (18.16) ∂t2 m ∂ x 2 which is the Schrödinger equation for a particle moving in one dimension acted on by a conservative force. The solution is in general a function of position and time. We might want to predict the momentum of a particle at a particular time. However, due to the probabilistic nature of quantum mechanics, of the momentum in identical systems will give a distribution of values. What can be done is to find the expectation value of the momentum

∞ ∞ ∂ψ p=∫ψ* p ψ dx = − iℏ ∫ ψ * dx . (18.17) −∞ −∞ ∂x Here it is assumed that the wave function is normalized so that

∞ ∫ ψ* ψ dx = 1. (18.18) −∞ A more general form of the Schrödinger equation is ∂ψ iℏ = H ψ , (18.19) ∂t

where H is the Hamiltonian operator. In many cases (but not all) the Hamiltonian operator is the same as that for the total energy. For a single particle acted on by conservative forces and viewed in an inertial reference frame,

ℏ2 H=− ∇+2 V ()r . (18.20) 2m The boundary conditions are that the wave function is zero on the boundary of the region accessible to the particle. This region might be all space, in which case the wave function must be zero at infinity.

The time independent Schrödinger equation If the Hamiltonian is time – independent, normal mode solutions of the Schrödinger equation can be found. We assume that the solution has time-dependence of form e−iω t , the same as for the plane wave. Equation (18.19) then gives ℏωψ= H ψ , (18.21) which, on recognizing that E = ℏω, is usually written as Hψ= E ψ . (18.22) This is the time independent Schrödinger equation. For a particle of mass m acted on by a conservative force, this equation is ℏ2 −∇+2ψV()r ψ = E ψ . (18.23) 2m