Introduction to Quantum Mechanics - Online Tutor, Practice Problems & Exam Prep

Quantum mechanics is a theoretical framework that describes the behavior of electrons at the atomic level. It employs quantum numbers, which are a set of values that specify the energy levels and positions of electrons within an atom. These numbers include the principal quantum number ($n$), which indicates the energy and size of an atomic shell; the angular momentum quantum number ($l$), also known as the azimuthal quantum number, which defines the shape of the orbital within a subshell; the magnetic quantum number ($m_l$), which provides the orientation of an orbital; and the spin quantum number ($m_s$), which determines the direction of an electron's spin within an orbital. Together, these quantum numbers allow for the precise identification of an electron's location within an atom, moving from a macro view of the atom down to the specific electron of interest. Understanding quantum numbers is essential for grasping the complex behavior of electrons, which in turn influences the chemical and physical properties of elements.

Introduction to Quantum Mechanics

Now quantum mechanics represents the mathematical or theoretical description of matter and its electrons on the atomic scale. It's a way of us describing the behavior of electrons within any given atom. Now if we were to take a mathematical approach to it, we'd have to use short Schrodinger's wave equation.

So if we take a look, we can see that this equation is pretty complicated. We have ψ in there, we have Δ. But don't worry, this type of equation isn't necessary for this level of chemistry. So we're going to bypass the mathematical approach in describing the behavior of electrons.

Instead, we're going to take more of a theoretical approach. And that means that we're going to have to understand the quantum numbers. Now, some people may hate the quantum numbers. We're going to go piece by piece and understand how the four quantum numbers themselves help to explain the location, and therefore the behavior of any particular electron within a given atom.

Introduction to Quantum Mechanics

Remember the quantum numbers are our theoretical approach in understanding the behavior of electrons. They represent a set of values that describe the energy levels and ultimately the location of a specific electron. Now within an atom we have what we first call our shell. The shell is the orbit that the electrons take as they travel around the nucleus.

Now within this shell we find what are called sub shells. These sub shells are the region where a group of electrons in an atom are located within the same shell. The sub shells use the variables of S, P, D, and F. Now within these subshells we have our orbitals. This is the region within a subshell where specific electrons can be found.

So if we take a look here at our quantum view, we have first our shell and within our shell we have our subshell. Here in blue and within our subshell we have our orbital in red and within a given orbital is where you find your electrons. These electrons spin will learn either clockwise or counterclockwise, so we'll see what that entails later on.

Now this quantum view connects directly to our quantum numbers. So with our quantum numbers we have our principal quantum number, our angular momentum quantum number, our azimuthal quantum number. So both terms are interchangeable. The magnetic quantum number and our spin quantum number. The principal quantum number deals with the energy and size of a shell, and it uses the variable N.

Next we have our angular momentum quantum number, also called our azimuthal quantum number. It deals with the shape of an orbital within a subshell. It uses the variable L. Next we have our magnetic quantum number. Here this just gives us the orientation of electrons in a set of orbitals, but more specifically it gives us the location of electrons within a set of orbitals. It uses the variable Ml.

And then finally we have our spin quantum number and from the name it tells us the spin of an electron in an orbital and it uses the variable Ms. Now all four quantum numbers together ultimately give us the location of an exact electron within an atom. Looking at all four quantum numbers together we start off first with the atom and within the atom we look at its shell, one particular shell.

Within that shell we find our sub shell. Within our sub shell we have our orbitals and within a specific orbital is where we locate finally the electron of interest. So this is the basic breakdown for the quantum numbers. We start off macro view looking at an atom overall and as we start looking at all the quantum numbers together, it gets us more and more and more specific in terms of an electron that we seek. All right. So that's the way we use the quantum numbers to understand the behavior of electrons and ultimately their locations.

Introduction to Quantum Mechanics Example 1

Now here it says if the path of an electron within an orbital can be seen as an... which quantum number would best describe this image? Alright, so they're saying that the electron is traveling in an... so an... Can be seen as something like this.

So they're talking about the specific shape that the electron create as it moves along its path. So we're talking about a shape and we know that if we're talking about a shape that has to deal with the angular momentum quantum number, another name for the angular momentum quantum number with BD as a muthel quantum number.

So that would mean that option B would be our correct answer.

Here’s what students ask on this topic:

What is quantum mechanics?

Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy at the atomic and subatomic levels. It departs from classical mechanics by introducing new principles, such as quantization of energy, wave-particle duality, and the uncertainty principle. In quantum mechanics, particles like electrons are described by wave functions, which provide probabilities of finding a particle in a particular place at a certain time, rather than definite positions. This theory has led to the development of various technologies, including semiconductors and lasers, and has profound implications for our understanding of the nature of reality. Quantum mechanics is essential in explaining phenomena that cannot be accounted for by classical physics, such as the behavior of atoms in a molecule, the properties of superconductors, and the stability of the electron's orbit in atoms.

How has quantum mechanics changed our understanding of atoms?

Quantum mechanics has revolutionized our understanding of atoms by introducing the concept that particles such as electrons exhibit both wave-like and particle-like properties. This dual nature is encapsulated in the wave-particle duality principle. Before quantum mechanics, atoms were thought to behave like tiny solar systems with electrons orbiting the nucleus in defined paths, known as the Bohr model. However, quantum mechanics introduced the probabilistic nature of these particles, leading to the electron cloud model where electrons are found in orbitals - regions of space where there is a high probability of finding an electron, rather than fixed orbits.

This framework also explains the quantization of energy levels within an atom, meaning electrons can only exist in specific energy states and can transition between these states by absorbing or emitting discrete amounts of energy, known as quanta. Quantum mechanics further allows for the understanding of chemical bonding, atomic stability, and the properties of elements based on their atomic structure, which are critical for explaining the behavior of matter at the atomic and subatomic levels.

What are quantum numbers?

Quantum numbers are a set of numerical values that describe the unique quantum state of an electron in an atom. They are crucial in understanding the arrangement of electrons in atoms, which is fundamental to the study of chemistry and physics. There are four quantum numbers:

1. Principal Quantum Number (n): This number determines the energy level or shell of an electron and is a positive integer (1, 2, 3, ...). The larger the number, the higher the energy level and the farther the electron is from the nucleus.
2. Angular Momentum Quantum Number (l): This number defines the shape of the electron's orbital and is an integer ranging from 0 to n-1. Each value of l corresponds to a specific orbital shape (s, p, d, f, etc.).
3. Magnetic Quantum Number (m_l): This number specifies the orientation of the orbital in space and can range from -l to +l, including zero. It determines the number of orbitals and their orientation within a subshell.
4. Spin Quantum Number (m_s): This number indicates the direction of the electron's spin and can either be +1/2 or -1/2, representing the two possible spin states of an electron.

Together, these quantum numbers allow us to predict the electron configuration of an atom, which is essential for understanding atomic structure, chemical bonding, and