new final exam: tuesday, may 8, 2012 starting at 8:30 a.m., hoyt...

Final Exam:

Tuesday, May 8, 2012 Starting at 8:30 a.m.,

Hoyt Hall.

Chapter 38 Quantum Mechanics

38-1 Quantum Mechanics – A New Theory

37-2 The Wave Function and Its Interpretation; the Double-Slit Experiment

38-3 The Heisenberg Uncertainty Principle

38-4 Philosophic Implications; Probability versus Determinism

38-5 The Schrödinger Equation in One Dimension – Time-Independent Form

38-6 Time-Dependent Schrödinger Equation

Units of Chapter 38

38-7 Free Particles; Plane Waves and Wave Packets

38-8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box)

38-9 Finite Potential Well

38-10 Tunneling Through a Barrier

Units of Chapter 38

pE c

⎛

⎝⎜

⎞

⎠⎟

k

ω c

⎛

⎝⎜

⎞

⎠⎟=

h2π

λ =hp

If waves can behave like particle, then particles can behave like waves

De Broglie wavelength

The Wave-Particle Duality

Electron diffraction

Photon diffraction

The Wave Nature of Matter

Figure 38.4

Double-Slit Experiment with Electrons

Imagine trying to see an electron with a powerful microscope. At least one photon must scatter off the electron and enter the microscope, but in doing so it will transfer some of i ts momentum to the electron.

38.3 The Heisenberg Uncertainty Principle

The combination of uncertainties gives:

which is called the

Heisenberg uncertainty principle.

It tells us that the position (x) and momentum (p) cannot simultaneously be measured with precision.

38.3 The Heisenberg Uncertainty Principle

(Δx) (Δpx ) ≥h2π

•  In Quantum Mechanics particles are represented by wave functions Ψ.

•  The absolute square of the wave function |Ψ|2 represents the probability of finding particle in a given location.

•  The wave function Ψ satisfies the

Time-Dependent Schrödinger Equation

Summary of Chapter 38 – Part I

−2

2m∂2

∂x2+V (x)

⎡

⎣⎢

⎤

⎦⎥Ψ(x,t) = i

∂∂t

Ψ(x,t)

One-Dimensional Wave Equation

∂2

∂x2D(x,t) = 1

v2∂2

∂t 2D(x,t)

−2

2m∂2

∂x2+V (x)

⎡

⎣⎢

⎤

⎦⎥Ψ(x,t) = i

∂∂t

Ψ(x,t)

One-Dimensional Schrödinger Equation

38.6 Time-Dependent Schrödinger Equation

Writing

we find that the Schrödinger equation can be separated, that is,

its time- and space-dependent parts can be solved for separately.

Ψ(x, t) =ψ (x) f (t)

−2

2m∂2

∂x2+V (x)

⎡

⎣⎢

⎤

⎦⎥Ψ(x,t) = i

∂∂t

Ψ(x,t)


Separation of the Time and Space Dependencies

Ψ(x, t) =ψ (x) f (t) →−

2

2m1

ψ (x)d 2

dx2ψ (x)+V (x) = E

i 1f (t)

ddtf (t) = E

⎧

⎨⎪⎪

⎩⎪⎪

38.6 Time-Dependent Schrödinger Equation

The time dependence can be found easily; it is:

This function has an absolute value of 1; it does not affect the probability density in space.

i 1f (t)

ddtf (t) = E → f (t) = e

−i E

⎛⎝⎜

⎞⎠⎟ t

−2

2m∂2

∂x2+V (x)

⎡

⎣⎢

⎤

⎦⎥Ψ(x,t) = i

∂∂t

Ψ(x,t)


−2

2md 2

dx2ψ (x) +V (x)ψ (x) = Eψ (x)

ddtf (t) = 1

iEf (t)

⎧

⎨⎪⎪

⎩⎪⎪

→ Ψ(x,t) =ψ (x)exp[−i Et]

Time-Independent Schrödinger Equation

38.5 The Schrödinger Equation in One Dimension—Time-Independent Form The Schrödinger equation cannot be derived, just as Newton’s laws cannot. However, we know that it must describe a traveling wave, and that energy must be conserved.

Therefore, the wave function will take the form:

where ψ(x) = Asinkx + Bcoskx

p = h2π

k = k → k = p

38.7 Free Particles; Plane Waves and Wave Packets

Free particle: no force, so V = 0.

The Schrödinger equation becomes:

− 2

2m∂2

∂x2ψ (x) = Eψ (x)

d 2

dx2ψ (x)+ 2mE

2ψ (x) = 0

which can be written


Free particle: no force, so V = 0.

d 2

dx2ψ (x)+ 2mE

2ψ (x) = 0

The solution to this equation is

with

ψ(x) = Asinkx + Bcoskx

k = 2mE2


Example 38-4: Free electron.

An electron with energy E = 6.3 eV is in free space (where V = 0). Find

(a) the wavelength λ (in nm) and

(b) the wave function for the electron (assuming B = 0).


The solution for a free particle is a plane wave, as shown in part (a) of the figure; more realistic is a wave packet, as shown in part (b). The wave packet h a s b o t h a r a n g e o f m o m e n t a a n d a f i n i t e uncertainty in width.

38.8 Particle in an Infinitely Deep Square Well Potential (a Rigid Box)

One of the few potentials where the Schrödinger equation can be solved exactly is the infinitely deep square well.

As is shown, this potential is zero from the origin to a distance , and is infinite elsewhere.


The solution for the region between the walls is that of a free particle:

Requiring that ψ = 0 at x = 0 and x =

gives B = 0 and k = nπ/ .

This means that the energy is limited to the values:

En = n2 h2

8ml2; n =1, 2, 3, ...


The wave function for each of the quantum states is:

ψ(x) = Asinknx ; kn = nπl

ψ(x) 2 dx−∞

+∞

∫ =1;

Asinknx2 dx

−∞

+∞

∫ = A 2 sinknx2 dx =

0

l

∫ 1;

The constant A is determined by the condition

ψn =2lsinknx


These plots show for several values of n the energy levels, wave function, and probability distribution.

energy levels wave function probability distribution


Example 38-5: Electron in an infinite potential well.

(a)  Calculate the three lowest energy levels for an electron trapped in an infinitely deep square well potential of width = 0.1 nm (about the diameter of a hydrogen atom in its ground state).

(b) If a photon were emitted when the electron jumps from the n = 2 state to the n = 1 state, what would its wavelength be?


Example 38-6: Calculating a normalization constant .

Show that the normalization constant A for all wave functions describing a particle in an infinite potential well of width has a value of:

2 / .A = l


Example 38-7: Probability near center of rigid box.

An electron is in an infinitely deep square well potential of width = 1.0 x 10-10 m. If the electron is in the ground state, what is the probability of finding it in a region dx = 1.0 x 10-12 m of width at the center of the well (at x = 0.5 x 10-10 m)?


Example 38-8: Probability of e- in ¼ of box.

Determine the probability of finding an electron in the left quarter of a rigid box—i.e., between one wall at x = 0 and position x = /4. Assume the electron is in the ground state.


Example 38-9: Most likely and average positions.

Two quantities that we often want to know are the most likely position of the particle and the average position of the particle. Consider the electron in the box of width

= 1.00 x 10-10 m in the first excited state n = 2.

(a)  What is its most likely position?

(b) What is its average position?


ψ (x) 2 = 2lsin2 2π

lx⎛

⎝⎜⎞⎠⎟

x = x0

l

∫ ψ (x) 2 dx


Example 38-10: A confined bacterium.

A tiny bacterium with a mass of about 10-14 kg is confined between two rigid walls 0.1 mm apart. (a) Estimate its minimum speed. (b) If instead, its speed is about 1 mm in 100 s, estimate the quantum number of its state.

38.9 Finite Potential Well A finite potential well has a potential of zero between x = 0 and x = , but outside that range the potential is a constant U0.

38.9 Finite Potential Well The potential outside the well is no longer zero; it falls off exponentially.

The wave function in the well is different than that outside it; we require that both the wave function and its first derivative be equal at the position of each wall (so it is continuous and smooth), and that the wave function be normalized.

38.9 Finite Potential Well

We require that both the wave function and its first derivative be equal at the position of each wall (so it is continuous and smooth), and that the wave function be normalized.

ψ II = Asin kx + Bcoskx

x = 0 x = l

ψ I = CeGx ψ III = De

−Gx

38.9 Finite Potential Well These graphs show the wave functions and probability distributions for the first three energy states.

Figure 38-13a goes here.

Figure 38-13b goes here.

Simulations

38.10 Tunneling Through a Barrier

Since the wave function does not go to zero immediately upon encountering a finite barrier, there is some probability of finding the particle represented by the wave function on the other side of the barrier. This is called tunneling.

Figure 38-15 goes here.


The probability that a particle tunnels through a barrier can be expressed as a transmission coefficient, T, and a reflection coefficient, R (where T + R = 1). If T is small,

The smaller E is with respect to U0, the smaller the probability that the particle will tunnel through the barrier.


Example 38-11: Barrier penetration.

A 50-eV electron approaches a square barrier 70 eV high and (a) 1.0 nm thick, (b) 0.10 nm thick. What is the probability that the electron will tunnel through?

38.10 Tunneling Through a Barrier Alpha decay is a tunneling process; this is why alpha decay lifetimes are so variable.

Scanning tunneling microscopes image the surface of a material by moving so as to keep the tunneling current constant. In doing so, they map an image of the surface. Figure 38-18 goes

here.

The 7.1 nm "quantum corral" shown above was created and imaged with an ultra-high vacuum cryogenic STM. Each sharp peak in the circle is an iron atom resting on atomically flat copper.


Macro scale image of an etched tungsten scanning tunneling microscopy (STM) tip. Uniform geometry contributes to a more stable tip.

Atomic Force Microscope

The enzyme-DNA complex can be imaged using AFM. It a l lows to reso lve the different stages of the restriction reaction: binding of the MTase and the motor subunits, grabbing of the D N A b y t h e m o t o r s , formation of the initial DNA loop, and the enzyme translocation.

Atomic Force Microscope

new final exam: tuesday, may 8, 2012 starting at 8:30 a.m., hoyt...

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