1 PHYS 3313 – Section 001 Lecture #18 Monday, Mar. 31, 2014 Dr. Jaehoon Yu Valid Wave Functions...

1

PHYS 3313 – Section 001Lecture #18

Monday, Mar. 31, 2014Dr. Jaehoon Yu

• Valid Wave Functions• Energy and Position Operators• Infinite Square Well Potential• Finite Square Well Potential

Monday, Mar. 31, 2014 PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu


2

Announcements• Research paper template has been uploaded onto

the class web page link to research• Special colloquium on April 2, triple extra credit• Colloquium this Wednesday at 4pm in SH101

Special Project #4• Prove that the wave function Ψ=A[sin(kx-ωt)

+icos(kx-ωt)] is a good solution for the time-dependent Schrödinger wave equation. Do NOT use the exponential expression of the wave function. (10 points)

• Determine whether or not the wave function Ψ=Ae-α|x| satisfy the time-dependent Schrödinger wave equation. (10 points)

• Due for this special project is Monday, Apr. 7.• You MUST have your own answers!

Monday, Mar. 31, 2014 3PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu

Properties of Valid Wave FunctionsBoundary conditions1) To avoid infinite probabilities, the wave function must be finite

everywhere.2) To avoid multiple values of the probability, the wave function must be

single valued.3) For finite potentials, the wave function and its derivatives must be

continuous. This is required because the second-order derivative term in the wave equation must be single valued. (There are exceptions to this rule when V is infinite.)

4) In order to normalize the wave functions, they must approach zero as x approaches infinity.

Solutions that do not satisfy these properties do not generally correspond to physically realizable circumstances.


Position and Energy Operators The position x is its own operator as seen above. The time derivative of the free-particle wave function

is

Substituting ω = E / ħ yields

The energy operator is The expectation value of the energy is


5

Infinite Square-Well Potential• The simplest such system is that of a particle trapped in a

box with infinitely hard walls that the particle cannot penetrate. This potential is called an infinite square well and is given by

• The wave function must be zero where the potential is infinite.

• Where the potential is zero inside the box, the time independent Schrödinger wave equation becomes where .

• The general solution is .Monday, Mar. 31, 2014 6PHYS 3313-001, Spring 2014 Dr. Jaehoon Yu

Quantization• Since the wave function must be continuous, the boundary conditions

of the potential dictate that the wave function must be zero at x = 0 and x = L. These yield valid solutions for B=0, and for integer values of n such that kL = nπ k=nπ/L

• The wave function is now

• We normalize the wave function

• The normalized wave function becomes

• These functions are identical to those obtained for a vibrating string with fixed ends.


Quantized Energy• The quantized wave number now becomes• Solving for the energy yields

• Note that the energy depends on the integer values of n. Hence the energy is quantized and nonzero.

• The special case of n = 1 is called the ground state energy.


How does this correspond to Classical Mech.?• What is the probability of finding a particle in a box of length L?• Bohr’s correspondence principle says that QM and CM must

correspond to each other! When?– When n becomes large, the QM approaches to CM

• So when n∞, the probability of finding a particle in a box of length L is

• Which is identical to the CM probability!!• One can also see this from the plot of P!


Determine the expectation values for x, x2, p and p2 of a particle in an infinite square well for the first excited state.

Ex 6.8: Expectation values inside a box


What is the wave function of the first excited state? n=? 2

A typical diameter of a nucleus is about 10-14m. Use the infinite square-well potential to calculate the transition energy from the first excited state to the ground state for a proton confined to the nucleus.

Ex 6.9: Proton Transition Energy


The energy of the state n is

n=1What is n for the ground state?

What is n for the 1st excited state? n=2

So the proton transition energy is

Finite Square-Well Potential• The finite square-well potential is

• The Schrödinger equation outside the finite well in regions I and III is

for regions I and III, or using

yields . The solution to this differential has exponentials of the form

eαx and e-αx. In the region x > L, we reject the positive exponential and

in the region x < 0, we reject the negative exponential. Why?


This is because the wave function should be 0 as xinfinity.

• Inside the square well, where the potential V is zero and the particle is free, the

wave equation becomes where

• Instead of a sinusoidal solution we can write

• The boundary conditions require that

and the wave function must be smooth where the regions meet.• Note that the

wave function is nonzero outside of the box.

• Non-zero at the boundary either..

• What would the energy look like?

Finite Square-Well Solution


Penetration Depth• The penetration depth is the distance outside the

potential well where the probability significantly decreases. It is given by

• It should not be surprising to find that the penetration distance that violates classical physics is proportional to Planck’s constant.


• The wave function must be a function of all three spatial coordinates.

• We begin with the conservation of energy• Multiply this by the wave function to get

• Now consider momentum as an operator acting on the wave function. In this case, the operator must act twice on each dimension. Given:

• The three dimensional Schrödinger wave equation is

Three-Dimensional Infinite-Potential Well


Rewrite

Consider a free particle inside a box with lengths L1, L2 and L3 along the x, y, and z axes, respectively, as shown in the Figure. The particle is constrained to be inside the box. Find the wave functions and energies. Then find the ground energy and wave function and the energy of the first excited state for a cube of sides L.



What are the boundary conditions for this situation?Particle is free, so x, y and z wave functions are independent from each other!Each wave function must be 0 at the wall! Inside the box, potential V is 0.

A reasonable solution is

Using the boundary condition

So the wave numbers are

Consider a free particle inside a box with lengths L1, L2 and L3 along the x, y, and z axes, respectively, as shown in Fire. The particle is constrained to be inside the box. Find the wave functions and energies. Then find the round energy and wave function and the energy of the first excited state for a cube of sides L.



The energy can be obtained through the Schrödinger equation

What is the ground state energy?

When are the energies the same for different combinations of ni?

E1,1,1 when n1=n2=n3=1, how much?

Degeneracy*• Analysis of the Schrödinger wave equation in three

dimensions introduces three quantum numbers that quantize the energy.

• A quantum state is degenerate when there is more than one wave function for a given energy.

• Degeneracy results from particular properties of the potential energy function that describes the system. A perturbation of the potential energy, such as the spin under a B field, can remove the degeneracy.


*Mirriam-webster: having two or more states or subdivisions having two or more states or subdivisions

The Simple Harmonic Oscillator• Simple harmonic oscillators describe many physical situations: springs, diatomic molecules

and atomic lattices.

• Consider the Taylor expansion of a potential function:

The minimum potential at x=x0, so dV/dx=0 and V1=0; and the zero potential V0=0, we have

Substituting this into the wave equation:

Let and which yields .


Parabolic Potential Well

• If the lowest energy level is zero, this violates the uncertainty principle.• The wave function solutions are where Hn(x) are Hermite

polynomials of order n.• In contrast to the particle in a box, where the oscillatory wave function is a sinusoidal

curve, in this case the oscillatory behavior is due to the polynomial, which dominates at small x. The exponential tail is provided by the Gaussian function, which dominates at large x.


Analysis of the Parabolic Potential Well

• The energy levels are given by

• The zero point energy is called the Heisenberg limit:

• Classically, the probability of finding the mass is greatest at the ends of motion’s range and smallest at the center (that is, proportional to the amount of time the mass spends at each position).

• Contrary to the classical one, the largest probability for this lowest energy state is for the particle to be at the center.


Barriers and Tunneling• Consider a particle of energy E approaching a potential barrier of height V0 and the

potential everywhere else is zero.• We will first consider the case when the energy is greater than the potential barrier.• In regions I and III the wave numbers are:

• In the barrier region we have


Reflection and Transmission• The wave function will consist of an incident wave, a reflected wave, and a

transmitted wave.• The potentials and the Schrödinger wave equation for the three regions are as

follows:

• The corresponding solutions are:

• As the wave moves from left to right, we can simplify the wave functions to:


Probability of Reflection and Transmission• The probability of the particles being reflected R or transmitted T is:

• The maximum kinetic energy of the photoelectrons depends on the value of the light frequency f and not on the intensity.

• Because the particles must be either reflected or transmitted we have: R + T = 1

• By applying the boundary conditions x → ±∞, x = 0, and x = L, we arrive at the transmission probability:

• When does the transmission probability become 1?


Tunneling• Now we consider the situation where classically the particle does not have enough energy to

surmount the potential barrier, E < V0.

• The quantum mechanical result, however, is one of the most remarkable features of modern physics, and there is ample experimental proof of its existence. There is a small, but finite, probability that the particle can penetrate the barrier and even emerge on the other side.

• The wave function in region II becomes

• The transmission probability that describes the phenomenon of tunneling is


Uncertainty Explanation• Consider when κL >> 1 then the transmission probability becomes:

• This violation allowed by the uncertainty principle is equal to the negative kinetic energy required! The particle is allowed by quantum mechanics and the uncertainty principle to penetrate into a classically forbidden region. The minimum such kinetic energy is:


Analogy with Wave Optics• If light passing through a glass prism reflects from an internal surface with an angle greater

than the critical angle, total internal reflection occurs. The electromagnetic field, however, is not exactly zero just outside the prism. Thus, if we bring another prism very close to the first one, experiments show that the electromagnetic wave (light) appears in the second prism.

• The situation is analogous to the tunneling described here. This effect was observed by Newton and can be demonstrated with two prisms and a laser. The intensity of the second light beam decreases exponentially as the distance between the two prisms increases.


Potential Well

• Consider a particle passing through a potential well region rather than through a potential barrier.

• Classically, the particle would speed up passing the well region, because K = mv2 / 2 = E - V0.According to quantum mechanics, reflection and transmission may occur, but the wavelength inside the potential well is shorter than outside. When the width of the potential well is precisely equal to half-integral or integral units of the wavelength, the reflected waves may be out of phase or in phase with the original wave, and cancellations or resonances may occur. The reflection/cancellation effects can lead to almost pure transmission or pure reflection for certain wavelengths. For example, at the second boundary (x = L) for a wave passing to the right, the wave may reflect and be out of phase with the incident wave. The effect would be a cancellation inside the well.


Alpha-Particle Decay• May nuclei heavier than Pb emits alpha particles (nucleus of He)! The phenomenon

of tunneling explains the alpha-particle decay of heavy, radioactive nuclei.• Inside the nucleus, an alpha particle feels the strong, short-range attractive nuclear

force as well as the repulsive Coulomb force.• The nuclear force dominates inside the nuclear radius where the potential is

approximately a square well.• The Coulomb force dominates

outside the nuclear radius.• The potential barrier at the nuclear

radius is several times greater than the energy of an alpha particle (~5MeV).

• According to quantum mechanics, however, the alpha particle can “tunnel” through the barrier. Hence this is observed as radioactive decay.


Application of the Schrödinger Equation to the Hydrogen Atom

• The approximation of the potential energy of the electron-proton system is the Coulomb potential:

• To solve this problem, we use the three-dimensional time-independent Schrödinger Equation.

• For Hydrogen-like atoms with one electron (He+ or Li++)• Replace e2 with Ze2 (Z is the atomic number)

• Use appropriate reduced mass μ


Application of the Schrödinger Equation The potential (central force) V(r) depends on the distance r

between the proton and electron.


• Transform to spherical polar coordinates to exploit the radial symmetry.

• Insert the Coulomb potential into the transformed Schrödinger equation.

Application of the Schrödinger Equation

• The wave function ψ is a function of r, θ and φ .The equation is separable into three equations of

independent variablesThe solution may be a product of three functions.

• We can separate the Schrodinger equation in polar coordinate into three separate differential equations, each depending only on one coordinate: r, θ, or φ .


Solution of the Schrödinger Equation for Hydrogen• Substitute ψ into the polar Schrodinger equation and separate the

resulting equation into three equations: R(r), f(θ), and g(φ).Separation of Variables• The derivatives in Schrodinger eq. can be written as

• Substituting them into the polar coord. Schrodinger Eq.

• Multiply both sides by r2 sin2 θ / Rfg


Reorganize

Solution of the Schrödinger Equation• Only r and θ appear on the left-hand side and only φ appears

on the right-hand side of the equation• The left-hand side of the equation cannot change as φ

changes.• The right-hand side cannot change with either r or θ.• Each side needs to be equal to a constant for the equation to

be true in all cases. Set the constant −mℓ2 equal to the right-

hand side of the reorganized equation

– The sign in this equation must be negative for a valid solution • It is convenient to choose a solution to be .

-------- azimuthal equation


Solution of the Schrödinger Equation• satisfies the previous equation for any value of mℓ.• The solution be single valued in order to have a valid solution for

any φ, which requires

• mℓ must be zero or an integer (positive or negative) for this to work• Now, set the remaining equation equal to −mℓ

2 and divide either side with sin2θ and rearrange them as

• Everything depends on r on the left side and θ on the right side of the equation.


Date post:	18-Jan-2016
Category:	Documents
Upload:	elvin-york
View:	213 times
Download:	0 times

1 PHYS 3313 – Section 001 Lecture #18 Monday, Mar. 31, 2014 Dr. Jaehoon Yu Valid Wave Functions...

Documents