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

Quantum Mechanics – A New Theory

In the early 1920s, it became increasingly evident that a new, more comprehensive theory was needed.

The new theory, called quantum mechanics, has been extremely successful in unifying into a single consistent theory

the wave-particle duality, black-body radiation, atoms, molecules, and many other phenomena. It is widely accepted as being the fundamental theory underlying all physical processes.

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

The properties of waves, such as interference and diffraction, are significant only when the size of objects or slits is not much larger than the wavelength. If the mass is really small, the wavelength can be large enough to be measured.

The Wave Nature of Matter

λ =hp

Electron diffraction

Photon diffraction

The Wave Nature of Matter

What is oscillating in a matter wave?

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

It is the probability of finding the particle that waves.

A matter wave is described by the wave function, Ψ. The square of the wave function |Ψ|2 (probability distribution) at any point is proportional to the number of particles expected to be found there. For a single particle, the wave function is the probability of finding the particle at that point.

The interference pattern is observed after many electrons have gone through the slits. If we send the electrons through one at a time, we cannot predict the path any single electron will take, but we can predict the overall distribution.

38.2 The Wave Function and Its Interpretation; the Double-Slit

Experiment

Figure 38.4

Double-Slit Experiment with Electrons

Double-Slit Experiment with Electrons

|Ψ| represents the matter wave amplitude and

|Ψ|2 represents the probability of finding a given electron at a given point.

Quantum mechanics tells us there are limits to measurement – not because of the limits of our instruments, but inherently.

This is due to wave-particle duality, and to interaction between the observing equipment and the object being observed.

38.3 The Heisenberg Uncertainty Principle

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 uncertainty in the momentum of the electron is taken to be the momentum of the photon – it could transfer anywhere from none to all of its momentum.

In addition, the position can only be measured to about one wavelength of the photon.

38.3 The Heisenberg Uncertainty Principle

Δpx ≈ h λ

Δx ≈ λ

The combination of uncertainties gives:

which is called the

Heisenberg uncertainty principle.

It tells us that the position (x) and momentum (p) cannot be measured with infinite precision at the same time.

38.3 The Heisenberg Uncertainty Principle

(Δx) (Δpx ) ≥h2π

The uncertainty principle applies also to time and energy:

This says that if an energy state only lasts for a limited time, its energy will be uncertain. It also says that conservation of energy can be violated if the interaction time is short enough.

38.3 The Heisenberg Uncertainty Principle

(ΔE) (Δt) ≥ h2π

The uncertainty principle applies also to angular variables

38.3 The Heisenberg Uncertainty Principle

(ΔLz ) (Δφ) ≥h2π

38.3 The Heisenberg Uncertainty Principle

(ΔLz ) (Δφ) ≥h2π

(ΔE) (Δt) ≥ h2π

(Δx) (Δpx ) ≥h2π

The uncer ta in ty p r inc ip le s ta tes a fundamental property of quantum systems, a n d i s n o t a s t a t e m e n t a b o u t t h e observational success of current technology.

The world of Newtonian mechanics is a deterministic one. If you know the forces on an object and its initial velocity, you can predict where it will go.

Quantum mechanics is very different – you can predict what ensembles of electrons will do, but have no idea what any individual one will do.

38.4 Philosophic Implications; Probability versus Determinism

38-6 The Time-Dependent Schrödinger Equation

What is the equation of motion of Ψ ?

D(x,t) = A sin(kx −ωt) → A ei(kx−ω t )

One-Dimensional Wave Equation ∂2D∂x2

−1v2

∂2D∂t 2

= 0

Harmonic Wave

∂2E∂x2

−1c2

∂2E∂t 2

= 01D EM Wave Equation in Vacuum:

What is the equation of motion of Ψ ?

38-6 The Time-Dependent Schrödinger Equation

What is the equation of motion of Ψ ?

−2

2m∂2

∂x2+V (x)

⎡

⎣⎢

⎤

⎦⎥Ψ(x,t) = i

∂∂t

Ψ(x,t)

This equation is satisfied by a harmonic wave function in the special case of a free particle (no net force acts)

V (x) = V0

so-called, Time-Dependent Schrödinger Equation

−2

2m∂2

∂x2+V0

⎡

⎣⎢

⎤

⎦⎥Ψ(x,t) = i

∂∂t

Ψ(x,t)

Ψ = A ei(kx−ω t )

2k2

2m+V0 = ω →

p2

2m+V0 = E

P(x,t) dx = Ψ 2 dx

The physical significance of the wave function Ψ is associated with the probability density

Ψ*Ψ = Ψ 2

The probability of finding a particle within a position range dx is

38-6 The Time-Dependent Schrödinger Equation

P(x, t) dV =1all space∫ one dimension Ψ 2 dx

−∞

∞

∫ =1

Normalization condition:

Since the solution to the Schrödinger equation is supposed to represent a single particle, the total probability of finding that particle anywhere in space should equal 1.

38-6 The Time-Dependent Schrödinger Equation

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