Quantum mechanics

Function. Analysis Model: Quantum Particle Under Boundary Conditions. The Schrödinger

Equation. A Particle in a Well of Finite Height. Tunneling Through a Potential Energy Barrier. Applications of Tunneling. The Simple Harmonic Oscillator.

that both matter and electromagnetic radiation are sometimes best modeled

as particles and sometimes as waves, depending on the phenomenon being observed. We can improve our understanding of quantum physics by making another connection between particles and waves using the notion of probability

We begin by discussing electromagnetic radiation using the particle model. The probability per unit volume of finding a photon in a given region of space at an instant of time is proportional to the number of photons per unit volume at that time:

proportional to the intensity of the radiation:
Now, let’s form a

connection between the particle model and the wave model by recalling that the intensity of electromagnetic radiation is proportional to the square of the electric field amplitude E for the electromagnetic wave:

Equating the beginning and the end of this series of proportionalities gives

with a particle is not a measurable quantity because the

wave function representing a particle is generally a complex function as we discuss below. In contrast, the electric field for an electromagnetic wave is a real function. The matter analog to Equation 41.1 relates the square of the amplitude of the wave to the probability per unit volume of finding the particle. Hence, the amplitude of the wave associated with the particle is called the probability amplitude, or the wave function, and it has the symbol Ψ.

first suggested by Max Born (1882–1970) in 1928. In 1926,

Erwin Schrцdinger proposed a wave equation that describes the manner in which the wave function changes in space and time.
The Schrцdinger represents a key element in the theory of quantum mechanics.

the particle will be found in the infinitesimal interval dx

around the point x is

position of a particle with complete certainty, it is possible

to specify the probability of observing it in a region surrounding a given point x. The probability of finding the particle in the arbitrary interval a ≤ x ≤ b is

x axis, the sum of the probabilities over all values

of x must be 1:

Any wave function satisfying this Equation is said to be normalized. Normalization is simply a statement that the particle exists at some point in space. Once the wave function for a particle is known, it is possible to calculate the average position at which you would expect to find the particle after many measurements. This average position is called the expectation value of x and is defined by the equation

any function f(x) associated with the particle by using the

following equation:

Example

Consider a particle whose wave function is graphed in Figure and is given by

(A) What is the value of A if this wave function is normalized?

particle?
The Wave Function
SOLUTION
Evaluate the expectation value:

particle’s momentum and energy. The quantum-mechanical approach to this problem

is quite different and requires that we find the appropriate wave function consistent with the conditions of the situation.

Analysis Model: Quantum Particle
Under Boundary Conditions

A Particle in a Box

particle in the box can be expressed as a real

sinusoidal function:

Therefore, only certain wavelengths for the particle are allowed!

A Particle in a Box

Equation gives
Normalizing this wave function shows that
Therefore, the normalized wave

function for the particle in a box is

A Particle in a Box

condition λ=2L/n, the magnitude of the momentum of the particle

is also restricted to specific values, which can be found from the expression for the de Broglie wavelength

Analysis Model: Quantum Particle
Under Boundary Conditions

We have chosen the potential energy of the system to be zero when the particle is inside the box. Therefore, the energy of the system is simply the kinetic energy of the particle and the allowed values are given by

A Particle in a Box

the particle in a box is not zero, then, according

to quantum mechanics, the particle can never be at rest! The smallest energy it can have, corresponding to n=1, is called the ground-state energy. This result contradicts the classical viewpoint, in which E=0 is an acceptable state, as are all positive values of E.

A Particle in a Box

General
In quantum mechanics, it is very common for particles to

be subject to boundary conditions. We therefore introduce a new analysis model, the quantum particle under boundary conditions. In many ways, this model is similar to the waves under boundary conditions model. In fact, the allowed wavelengths for the wave function of a particle in a box are identical in form to the allowed wavelengths for mechanical waves on a string fixed at both ends.
The quantum particle under boundary conditions model differs in some ways from the waves under boundary conditions model:
• In most cases of quantum particles, the wave function is not a simple sinusoidal function like the wave function for waves on strings. Furthermore, the wave function for a quantum particle may be a complex function.
• For a quantum particle, frequency is related to energy through E =hf, so the quantized frequencies lead to quantized energies.
• There may be no stationary “nodes” associated with the wave function of a quantum particle under boundary conditions. Systems more complicated than the particle in a box have more complicated wave functions, and some boundary conditions may not lead to zeroes of the wave function at fixed points.

General
In general,

an interaction of a quantum particle with its

environment represents one or more boundary conditions, and, if the interaction restricts the particle to a finite region of space, results in quantization of the energy of the system.

radiation that follows from Maxwell’s equations. The waves associated with

particles also satisfy a wave equation. The wave equation for material particles is different from that associated with photons because material particles have a nonzero rest energy. The appropriate wave equation was developed by Schrödinger in 1926. In analyzing the behavior of a quantum system, the approach is to determine a solution to this equation and then apply the appropriate boundary conditions to the solution. This process yields the allowed wave functions and energy levels of the system under consideration. Proper manipulation of the wave function then enables one to calculate all measurable features of the system.

particle of mass m confined to moving along the x

axis and interacting with its environment through a potential energy function U(x) is

where E is a constant equal to the total energy of the system (the particle and its environment). Because this equation is independent of time, it is commonly referred to as the time-independent Schrödinger equation.

shape of the curve in Figure, the particle in a

box is sometimes said to be in a square well, where a well is an upward-facing region of the curve in a potential-energy diagram.

determined by the boundary and normalization conditions.
everywhere, which is not

a valid wave function

a quantized energy that we call En. Solving for the

allowed energies En gives

Substituting the values of k in the wave function, the allowed wave functions are given by

where U=0
For regions I and III may be written
Because U>E,

the coefficient of c on the right-hand side is necessarily positive. Therefore, we can express Equation as

complete wave function, we impose the following boundary conditions:

this shape is called a square barrier, and U is

called the barrier height.

According to the uncertainty principle, the particle could be within the barrier as long as the time interval during which it is in the barrier is short. If the barrier is relatively narrow, this short time interval can allow the particle to pass through the barrier.

The movement of the particle to the far side of the barrier is called tunneling or barrier penetration.

be described with a transmission coefficient T and a reflection

coefficient R. The transmission coefficient represents the probability that the particle penetrates to the other side of the barrier, and the reflection coefficient is the probability that the particle is reflected by the barrier.

This quantum model of barrier penetration shows that T can be nonzero. That the phenomenon of tunneling is observed experimentally provides further confidence in the principles of quantum physics.

the classical model, any value of E is allowed, including

E=0, which is the total energy when the particle is at rest at x=0.

where the angular frequency of vibration is

is treated from a quantum point of view. The Schrödinger

equation for this problem is

have guessed corresponds to the ground state of the system,

which has an energy

The energy levels of a harmonic oscillator are quantized as we would expect because the oscillating particle is bound to stay near x=0. The energy of a state having an arbitrary quantum number n is