The 3D quantum harmonic oscillator

Summary

Professor MDA explores the three-dimensional quantum harmonic oscillator, a fundamental concept in quantum mechanics. The video explains how the system's properties can be studied using tensor products, simplifying the Hamiltonian into a sum of one-dimensional oscillators. It covers eigenvalues, eigenstates, ladder operators, and wave functions, emphasizing the oscillator's solutions are derived from combining one-dimensional counterparts.

Takeaways

• 🔬 The three-dimensional quantum harmonic oscillator is a fundamental concept in quantum mechanics that helps understand the behavior of atoms in solids and light.
• 📚 The script discusses the quantum harmonic oscillator from a new perspective, focusing on the transition from one to three dimensions using tensor products.
• 🧮 Tensor products are crucial for understanding the state space of a particle moving in three dimensions, represented as a combination of state spaces in each spatial dimension.
• 📐 The Hamiltonian for the three-dimensional quantum harmonic oscillator includes kinetic and potential energy terms for each of the x, y, and z directions.
• 🌐 The script emphasizes the importance of understanding tensor product state spaces to avoid ambiguities in quantum calculations.
• 🔑 The eigenvalue equation for the three-dimensional quantum harmonic oscillator can be solved by combining the solutions from one-dimensional problems.
• 📈 Eigenvalues of the three-dimensional quantum harmonic oscillator are quantized and are the sum of the eigenvalues from each of the three dimensions.
• 📉 Ladder operators are introduced as a tool to build eigenstates and understand the energy transitions in the quantum harmonic oscillator.
• 🌌 The eigenstates of the three-dimensional quantum harmonic oscillator are constructed as tensor products of the eigenstates from each one-dimensional direction.
• 🌟 The wave function of the three-dimensional quantum harmonic oscillator is the product of the wave functions of one-dimensional oscillators along each axis.
• 📝 The script provides a clear summary of how the solutions to the three-dimensional quantum harmonic oscillator can be derived from one-dimensional solutions, highlighting the simplicity and power of this approach.

Q & A

What is the main topic discussed in the video?

The main topic discussed in the video is the three-dimensional quantum harmonic oscillator, exploring its properties and solutions by extending the concepts from the one-dimensional quantum harmonic oscillator.

Why are tensor products important in quantum mechanics?

Tensor products are important in quantum mechanics because they allow for the description of quantum systems with multiple degrees of freedom, such as a particle moving in three spatial dimensions.

What is the Hamiltonian of a three-dimensional quantum harmonic oscillator?

The Hamiltonian of a three-dimensional quantum harmonic oscillator includes kinetic energy terms proportional to the momentum squared along the x, y, and z directions, and potential energy terms that depend on quadratic terms in x, y, and z.

How does the Hamiltonian for a three-dimensional quantum harmonic oscillator relate to the Hamiltonians of one-dimensional oscillators?

The Hamiltonian for a three-dimensional quantum harmonic oscillator can be expressed as the sum of the Hamiltonians for one-dimensional oscillators along each of the x, y, and z axes.

What are the eigenvalues of the three-dimensional quantum harmonic oscillator?

The eigenvalues of the three-dimensional quantum harmonic oscillator are given by the sum of the eigenvalues of the one-dimensional harmonic oscillators along the x, y, and z directions.

How are the eigenstates of the three-dimensional quantum harmonic oscillator constructed?

The eigenstates of the three-dimensional quantum harmonic oscillator are constructed by taking the tensor product of the eigenstates of the one-dimensional harmonic oscillators along each of the x, y, and z axes.

What role do ladder operators play in the study of the quantum harmonic oscillator?

Ladder operators are used to lower or raise the energy of a quantum state by one quantum of energy. They are essential for determining the allowed eigenvalues and for constructing the eigenstates of the quantum harmonic oscillator.

How are the wave functions of the three-dimensional quantum harmonic oscillator related to those of one-dimensional oscillators?

The wave function of the energy eigenstates of the three-dimensional quantum harmonic oscillator is the product of the wave functions of three one-dimensional harmonic oscillators along the x, y, and z axes.

What is the significance of being able to separate the Hamiltonian into components that act non-trivially along each spatial dimension?

The ability to separate the Hamiltonian into components that act non-trivially along each spatial dimension allows for the simplification of calculations and the combination of solutions from one-dimensional problems to solve the three-dimensional problem.

Why is it important to remember that we are working in a tensor product state space?

It is important to remember that we are working in a tensor product state space to avoid potential ambiguities and to correctly apply the properties of tensor products when solving problems in quantum mechanics.

What are some of the interesting properties that emerge from studying the three-dimensional quantum harmonic oscillator?

Some interesting properties that emerge from studying the three-dimensional quantum harmonic oscillator include degeneracies and the behavior of the system in an isotropic central potential.

Outlines

00:00

📚 Introduction to the 3D Quantum Harmonic Oscillator

Professor MDA begins the lecture by introducing the three-dimensional quantum harmonic oscillator, a fundamental concept in quantum mechanics. This oscillator is crucial for understanding the low-energy behavior of various systems, such as atomic motion in solids and light behavior. The lecture builds upon previous discussions on the one-dimensional quantum harmonic oscillator and coherent states. The focus of this video is to explore the implications of extending the concept from one to three dimensions, using tensor products to understand the state space of a particle in three spatial dimensions. The Hamiltonian of the system, which includes kinetic and potential energy terms, is introduced, and the video sets the stage for a deeper exploration of tensor products and their applications in quantum systems.

05:01

🔍 Decomposing the Hamiltonian in 3D Quantum Harmonic Oscillator

This section delves into the mathematical representation of the Hamiltonian for a three-dimensional quantum harmonic oscillator. The Hamiltonian is decomposed into separate components that act on individual spatial dimensions (x, y, z), highlighting the system's separability. Each component is a sum of kinetic and potential energy terms specific to its dimension. The lecture explains how the full Hamiltonian can be expressed as a sum of individual one-dimensional Hamiltonians, each corresponding to motion along a single axis. This simplification allows for the construction of the full solution by combining solutions from one-dimensional problems, emphasizing the power of tensor product state spaces in simplifying complex quantum systems.

10:02

🚀 Constructing Eigenstates and Eigenvalues in 3D Quantum Harmonic Oscillator

The lecture continues by discussing the eigenvalue equation for the three-dimensional quantum harmonic oscillator. It explains how the eigenvalues and eigenstates can be constructed by considering the motion along individual spatial directions. The eigenvalues are quantized and are the sum of the eigenvalues from each one-dimensional direction. The eigenstates are constructed as tensor products of the eigenstates from each direction. The section also introduces ladder operators, which are essential for understanding the action on eigenstates and for constructing the eigenstates of the system. The lecture provides a comprehensive overview of how the solutions to the one-dimensional problems can be combined to solve the three-dimensional problem.

15:02

🌊 Wave Functions and Simplifying Notations in Quantum Mechanics

This part of the lecture focuses on the wave functions of the three-dimensional quantum harmonic oscillator. It explains how the wave function can be constructed from the product of wave functions of one-dimensional oscillators along each axis. The lecture also discusses various notational simplifications used in quantum mechanics, such as omitting tensor product symbols and sub-indices, to make the representation more concise and manageable. The importance of understanding the underlying tensor product states despite these simplifications is emphasized. The section provides a clear explanation of how the wave functions of the three-dimensional oscillator relate to those of the one-dimensional oscillators, highlighting the simplicity and elegance of quantum mechanical solutions.

20:04

🔗 Summary of the 3D Quantum Harmonic Oscillator

The final section of the lecture summarizes the key points discussed about the three-dimensional quantum harmonic oscillator. It reiterates that the Hamiltonian of the 3D oscillator is a sum of three one-dimensional Hamiltonians, each corresponding to motion along the x, y, and z axes. The energy eigenvalues are the sum of the eigenvalues from each one-dimensional direction, and the eigenstates are the tensor products of the eigenstates from each direction. The wave function of the 3D oscillator is the product of the wave functions of the one-dimensional oscillators. The lecture concludes by encouraging viewers to explore further topics related to the three-dimensional quantum harmonic oscillator, such as its degeneracies and central potential, to gain a deeper understanding of quantum mechanical systems.

Keywords

💡Quantum Harmonic Oscillator

A quantum harmonic oscillator is a theoretical model used in quantum mechanics to describe the behavior of a particle in a potential energy environment that is proportional to the square of its displacement from a central point. In the video, the quantum harmonic oscillator is used to study systems ranging from atomic motion in solids to the behavior of light. The script discusses how the properties of a three-dimensional quantum harmonic oscillator can be understood by extending the principles of a one-dimensional model.

💡Tensor Products

Tensor products are mathematical constructs used to combine vectors from different vector spaces into a single vector. They are crucial in quantum mechanics for describing systems with multiple degrees of freedom, such as a particle moving in three dimensions. The script emphasizes the use of tensor products to simplify the Hamiltonian of a three-dimensional quantum harmonic oscillator by treating it as a combination of one-dimensional systems.

💡Hamiltonian

The Hamiltonian in quantum mechanics is an operator corresponding to the total energy of a system. It is used to describe the time evolution of a quantum system and is central to solving for the system's energy eigenvalues and eigenstates. In the video, the Hamiltonian for a three-dimensional quantum harmonic oscillator is detailed, showing how it is composed of kinetic and potential energy terms along each spatial dimension.

💡Eigenvalues and Eigenstates

Eigenvalues and eigenstates are fundamental concepts in linear algebra and quantum mechanics. An eigenstate of an operator is a state that, when acted upon by the operator, yields the same state multiplied by a scalar (the eigenvalue). The video script explains how the eigenvalues and eigenstates of the three-dimensional quantum harmonic oscillator are derived from those of one-dimensional systems, showcasing the separability of the Hamiltonian.

💡Coherent States

Coherent states are a special set of quantum states that have properties analogous to classical harmonic oscillator states. They provide a quasi-classical view of quantum systems. Although not directly discussed in the script, they are mentioned as a related topic that the viewer might explore for further understanding of the quantum harmonic oscillator.

💡Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion. In the context of the quantum harmonic oscillator, the kinetic energy term in the Hamiltonian is proportional to the square of the particle's momentum. The script describes how these terms contribute to the overall energy of the system and how they are separated into components along each spatial dimension.

💡Potential Energy

Potential energy is the energy stored in an object due to its position in a force field. For the quantum harmonic oscillator, the potential energy is proportional to the square of the displacement from the equilibrium position. The script explains how potential energy terms are included in the Hamiltonian and how they act on the state space.

💡Ladder Operators

Ladder operators are a pair of operators in quantum mechanics that allow for the systematic raising and lowering of energy levels in a system. The script mentions the lowering (a) and raising (a+) operators in the context of the quantum harmonic oscillator, explaining how they are used to construct eigenstates and understand energy transitions.

💡Position Representation

Position representation, or wave function, is a way of describing quantum states in terms of spatial coordinates. The script discusses how the wave functions of the three-dimensional quantum harmonic oscillator can be constructed from the wave functions of one-dimensional oscillators along each axis, emphasizing the product form of these wave functions.

💡Hermite Polynomials

Hermite polynomials are a set of orthogonal polynomials that play a crucial role in quantum mechanics, particularly in the description of the wave functions of the quantum harmonic oscillator. The script mentions that the wave functions of the quantum harmonic oscillator are composed of Hermite polynomials multiplied by a Gaussian function, which reflects the oscillator's Gaussian potential well.

Highlights

Introduction to the three-dimensional quantum harmonic oscillator and its importance in various scientific fields.

Explanation of how the quantum harmonic oscillator can be used to study properties of systems like atomic motion in solids and light behavior.

Discussion on the use of tensor products to understand the state space of a particle moving in three spatial dimensions.

The Hamiltonian of the three-dimensional quantum harmonic oscillator includes kinetic and potential energy terms for each spatial direction.

Simplification of the Hamiltonian using tensor product properties and the separation of terms acting on different state spaces.

The eigenvalue equation for the three-dimensional quantum harmonic oscillator is presented.

Solution of the eigenvalue equation by combining solutions of individual one-dimensional problems.

Quantization of eigenvalues and their dependence on non-negative integers for each spatial direction.

Construction of eigenstates using tensor products of eigenstates from individual spatial dimensions.

Introduction to ladder operators and their role in the quantum harmonic oscillator.

Description of how ladder operators can be extended to act on the full state space of the three-dimensional oscillator.

Derivation of the eigenstates of the three-dimensional quantum harmonic oscillator using ladder operators.

Explanation of the position representation and wave functions of the three-dimensional quantum harmonic oscillator.

Construction of the wave function by combining the wave functions of one-dimensional oscillators along each spatial direction.

Summary of the Hamiltonian's form and how it allows for easy construction of energy eigenvalues and eigenstates.

Discussion on the degeneracies of the three-dimensional quantum harmonic oscillator and its implications.

Encouragement to explore further properties of the isotropic three-dimensional quantum harmonic oscillator.

Transcripts

hi everyone this is professor mda

science and today i want to discuss the

three-dimensional quantum harmonic

oscillator in another one of our videos

on rigorous quantum mechanics harmonic

oscillations permeate science as the low

energy behavior of many systems is

harmonic for example the quantum

harmonic oscillator allows us to study

the properties of systems ranging from

the motion of atoms in solids to the

behavior of light we have a whole series

of videos on the one-dimensional quantum

harmonic oscillator where we learn how

to calculate its eigenvalues and

eigenstates and we also have a whole

series on coherent states which provide

a quasi-classical view of the quantum

harmonic oscillator today we want to

discuss the quantum family oscillator

from yet another point of view which is

what happens when we go from one to

three dimensions to do so we're going to

exploit the properties of tensor

products this means that this video is

also a great resource for you to

practice tensor products which is really

useful because they feature in many many

different quantum systems so let's go

let's consider a particle moving in a

three-dimensional quantum harmonic

oscillator

we call the state space of a particle

moving in three spatial dimensions v

and it's given by the tensor product of

the state spaces phi x v y and v z where

v x is the state space of a particle

moving in the x spatial dimension and

similarly for v y and v z

so as you can see to understand the

motion of particles in three spatial

dimensions we need to use the properties

of tensor product state spaces and if

you haven't seen the corresponding

videos yet i recommend that you check

them out first and continue with this

one afterwards

the hamiltonian h includes the kinetic

energy which is given by a term

proportional to the momentum squared

along the x direction a term

proportional to the momentum squared

along the y direction and the term

proportional to the momentum squared

along the z direction

and then we also have the potential

energy which depends on a quadratic term

in x

a quadratic term in y

and a quadratic term in z

the hamiltonian h acts on the full state

space v

and each of the terms in which we write

the hamiltonian also act on the full

state space v

however we see that the form that the

various terms take is rather simple

for example let's consider the first

term here

it is the kinetic energy of the particle

associated with momentum in the x

direction

this kinetic energy operator acts on the

full state space v but its action can be

separated into this part which acts on

vx only

this part which acts trivially on v y as

it is simply the corresponding identity

operator

and this part which again acts trivially

as the identity operator but now on vz

the only non-trivial part in this term

is the one acting on vx so we typically

simplify our notation to rewrite it like

this omitting the identity operators and

tensor product symbols

all other kinetic and potential energy

terms have a similar form with multiple

trivial parts so using the same

simplified notation we can write the

full hamiltonian as equal to the kinetic

energy along x

the kinetic energy along y and the

kinetic energy alongside

and then the potential energy along x

the potential energy along y and the

potential energy alongside

when we work with tensor product state

spaces we use this simple notation

whenever possible and if you've worked

with particles moving in three spatial

dimensions before it's very likely that

you'll have directly worked with this

latter simpler expression without

reference to the full original

expression up here

while this works fine for many types of

calculation it is important to keep in

mind that we're really working in a

tensor product state space as this can

become important to avoid potential

ambiguities

now to make sure that we become

comfortable with tensor product state

spaces we will combine the use of both

notations throughout the video

we can actually rewrite this hamiltonian

in another form that will prove really

convenient let's consider h

we can next group all the terms that act

non-trivially along the x direction

which are this kinetic energy term and

this potential energy term to end up

with this combined x-dependent term

and then the identity operators for the

y and z-directions

we can do the same for the terms acting

non-trivially along y to end up with

this identity

this y dependent term

and this other identity

and we can of course do the same for the

terms acting non-trivially along z to

end up with this identity this identity

and this z-dependent term

we now recognize this here as the

quantum harmonic oscillator hamiltonian

of a particle moving in the x-spatial

dimension

this here has a corresponding

hamiltonian of a particle moving in the

y spatial dimension and this here as a

corresponding hamiltonian of a particle

moving in the z spatial dimension

with this we can rewrite the hamiltonian

of a particle moving in a

three-dimensional quantum harmonic

oscillator as the sum of a term that

only involves the hamiltonian of the

particle moving in the x-direction

plus the hamiltonian of the particle

moving along y

plus the hamiltonian of the particle

moving along z

using the simplified notation we

discussed in the previous slide we can

rewrite this as the sum of h x plus h y

plus h z

so what have we accomplished

this last expression shows that the

hamiltonian h of a particle moving in a

three-dimensional quantum harmonic

oscillator is simply given by the sum of

the hamiltonians corresponding to the

particle moving in each of the three

dimensions separately

and this suggests that we will be able

to construct the full solution of the

three-dimensional problem simply by

combining the solutions of the

individual one-dimensional problems and

this is indeed what we're going to do in

the rest of this video

as always the solution of the quantum

harmonic oscillator involves the

solution of the eigenvalue equation

remember that for the three-dimensional

harmonic oscillator we work in the state

space v

and we write the eigenvalue equation as

h acting on psi

equal to e psi

where as usual these are the eigenvalues

and these are the eigen

states given the form of the hamiltonian

it will be useful to consider the

eigenvalue equations of the motion of

the particle along the individual

spatial directions

if we start with the state space vx of a

particle moving in the one-dimensional x

spatial direction then we have this

eigenvalue equation for the quantum

harmonic oscillator eigenvalue

we've already solved this problem of a

one-dimensional quantum harmonic

oscillator and from the corresponding

videos we know that the eigenvalues are

quantized and given by this expression

where nx is a non-negative integer

we can similarly work in state space v y

of a particle moving in the

one-dimensional y-spatial direction with

this eigenvalue equation

these quantized eigenvalues

and again the n y are non-negative

integers

and for v z we have the corresponding

eigenvalue equation

with the corresponding quantized

eigenvalues and the nz are still

non-negative integers

from the video on eigenvalues and

eigenstates of tensor product state

spaces linked in the description we know

that given the form of the hamiltonian

up here we can build the eigenvalues and

eigenstates of h from those of hx hy and

z

in particular the eigenstate psi is

given by the tensor product of the

eigenstates nx and y and nz

and the eigenvalue e

is given by the sum of the eigenvalues e

x e y and dz

and this is it we've solved the

eigenvalue equation of the

three-dimensional quantum harmonic

oscillator by simply using our knowledge

of the solution of the one-dimensional

quantum harmonic oscillator

in the rest of the video we will explore

some interesting features of the

three-dimensional quantum harmonic

oscillator

let's first consider the eigenvalues

remember that we've just figured out

that the eigenvalues e of the

three-dimensional quantum harmonic

oscillator are given by the sum of e n x

e and y and e and z

given the expressions up here we can

rewrite to the eigenvalue e as equal to

the sum of this term proportional to

omega x

this term proportional to omega y and

this term proportional to omega z

the eigenvalues of the three-dimensional

harmonic oscillator are labeled by a

collection of three numbers n x and y

and z

and this means that we can label the

distinct eigenvalues of the

three-dimensional quantum harmonic

oscillator with these three numbers

where nx and y and nz can each take any

non-negative integer value

to study the eigenstates it will first

prove convenient to consider the ladder

operators

let's start with vx from our videos on

the one-dimensional harmonic oscillator

we know that the lowering operator ax is

defined as this pre-factor

times the position operator plus this

other prefactor

times the momentum operator

the raising operator is the at joint a

dagger of the lowering operator and is

given by the corresponding term

proportional to position and the

negative of the corresponding term

proportional to momentum

these ladder operators are used

thoroughly in the study of the quantum

harmonic oscillator for example we use

them to figure out the allowed

eigenvalues but now we are interested in

their action on the eigenstates

the lowering operator acting on an

energy eigenstate gives another energy

eigenstate where we've removed one

quantum of energy

conversely the racing operator acting on

the same energy eigenstate gives another

energy eigenstate where we've added one

quantum of energy

this is a quick refresher on ladder

operators and for a full description of

these ideas you should check out the

videos in the description

for our purposes today the important

thing is that when we work in v y we can

define an analogous lowering operator

and analogous raising operator

and when we work in visit we can define

an analogous lowering operator

and an analogous racing operator as well

from the video on tensor product state

spaces we know how to extend the action

of these ladder operators to the full

state space

as an example consider the operator ax

identity y identity z

acting on an energy eigenstate n x n y

and z

from the video on tensor products we

know that each operator acts only on the

state from its original space so ax acts

on nx to get this

then the identity acts on ny to

trivially get ny back

and finally this identity acts on nz to

trivially get nz back

in the simplified notation for apparatus

we would simply write this as ax acting

on the tensor product state

giving this new tensor product state

although we only write a x we implicitly

understand that we also have the

identities in v y and v z

so again the simplified notation is more

convenient but we need to be sure that

we know what we're doing

the ladder operators are important

because we can use them to build the

eigen states

in vx we have that the eigen state nx is

equal to this prefactor

times the application of the raising

operator nx times

on the ground state

now remember that the ground state is

the state associated with the lowest

energy eigenvalue and is such that the

action of ax on it kills the state

in vy the energy eigenstate ny is given

by the corresponding action of the

rating operator on the ground state and

in vz the energy eigenstate nz is also

obtained by the application of the

corresponding raising operator on the

ground state

if we now move to the center product

state space v of the three-dimensional

harmonic oscillator we've determined

earlier that the eigenstates are given

by the tensor product of the eigenstates

in v x v y

and v z

now the expression for this eigenstate

is rather long so bear with me

using these expressions up here we first

get this combined pre-factor

then the application of ax a total of nx

times the application of a y a total of

n y times and the application of azit

for a total of nz times

all of this acting on the ground state

of the three-dimensional oscillator

and these are the eigen states of the

three-dimensional quantum harmonic

oscillator as built from the rating

operators

just like we do for operators we can use

simpler notation to describe tensor

product states such as these

let's take the eigenstate nx and y and z

a common simplification is the emission

of the tensor product symbol

another simplification typically used is

the emission of the sub-indices

indicating the original state space to

which the states belong to

in this case the order of the terms

indicates the state space of origin in

other words we understand that the first

kit comes from vx the second from v y

and the third from v z

yes another simplification is to group

these together into a single kit where

the various labels are separated by

commas

you'll encounter all of these

conventions and again it is essential

that you always remember what you are

really dealing with tensor product

states

using this simple notation together with

the simple notation for operators we can

rewrite the eigenstates of the

three-dimensional quantum harmonics

later as equal to this pre-factor

times the action of ax that of a y and

that of a z on the ground state

sticking with eigenstates we're now

going to look at the position

representation or to put it another way

at the wave functions of the

three-dimensional quantum harmonic

oscillator if we start in the vx state

space remember that for a one

dimensional harmonic oscillator the wave

function associated with energy

eigenstate nx is labeled as psi nx of x

and is given by the usual bracket

between the position eigen states and

the energy eigenstate we are interested

in

in the video on the quantum harmonic

oscillator eigenstates we show that this

wave function can be written as this

prefactor

multiplied by a polynomial of order in x

called a hermit polynomial multiplied by

a gaussian

as you can imagine if we look at v y we

have an analogous expression for the

wave function in terms of the

corresponding prefactor

the hermit polynomial and the gaussian

and if we look at vz we also get the

wave function again as equal to this

prefactor the hermit polynomial and the

gaussian

we can now look at the state space v of

the three-dimensional oscillator

we're going to call the wave function

psi n x and y and z and it is a function

of the three variables x y z

it is given by the bracket between the

position basis states

and the energy eigenstates that we're

interested in

from the video on tensor products we

know that we can calculate this bracket

by grouping together the objects in the

same state spaces making up the tensor

product state space

in vx we get this bracket we then need

to multiply this by the corresponding

bracket in v y and by the corresponding

bracket in v z

we can now construct the full wave

function by using the wave functions

along the individual one-dimensional

spaces above

for the scalar products in vx we get the

psi and x wave function for the scalar

product of v y we get the psi and y wave

function and for the scalar products in

v z we get the psi and z wave function

and this is it

this is the wave function of the energy

eigenstates of the three-dimensional

quantum harmonic oscillator

it is simply the product of the wave

function of a one-dimensional oscillator

along x with a one-dimensional

oscillator along y

with a one-dimensional oscillator along

z

now a word of caution

notation could be somewhat confusing so

let's make sure that we really do

understand it

when we write psi here with three

sub-indices we mean the wave function of

the three-dimensional quantum harmonic

oscillator

when we write psi with a single

sub-index for example this one here we

mean the wave function of the

one-dimensional harmonic oscillator and

in this case it corresponds to a

one-dimensional harmonic oscillator

along the z-axis

okay so these are the wave functions

for the one-dimensional harmonic

oscillator we spend some time plotting

them

unfortunately it isn't possible to fully

plot the eigenfunctions of the three

dimensionals later as we would need a

four-dimensional space to do so

but we could certainly visualize them

along the different spatial axes where

they essentially look like they're

one-dimensional counterparts so i

encourage you to check out the video on

the eigenfunctions of the

one-dimensional quantum harmonic

oscillator for a reminder of what they

look like

let's finish with a brief summary this

is the hamiltonian of a

three-dimensional quantum harmonic

oscillator

in this expression i spell it out in its

full glory in terms of tensor products

of operators associated with one

dimensional harmonic oscillators along

the x y and z directions

for example this term here is a term

that contains a kinetic energy of the

particle moving in the x direction and

only acts trivially along the y and z

directions

when we work with tensor product state

spaces we tend to simplify our notation

if we can do so without introducing any

ambiguity

in this case the simple notation is

given by this hamiltonian down here

the hamiltonians in either notation show

that the three-dimensional quantum

harmonic oscillator is essentially a

simple sum of three hamiltonians

corresponding to the separate

one-dimensional motion of the particle

in each of the x y and z dimensions

we've seen that this simple form for the

hamiltonian means that it is very easy

to construct the energy eigenvalues and

eigen states of the three-dimensional

harmonic oscillator from those of the

one-dimensional oscillator

the energy eigenvalues of the

three-dimensional quantum harmonic

oscillator are simply given by the sum

of the energy eigenvalues of a

one-dimensional harmonic oscillator

along the x-direction the y-direction

and the z-direction

this longer expression shows the

explicit form of these energy

eigenvalues

we can also easily build the energy's

eigenstates they are simply given by the

tensor products of the energy

eigenstates of the individual

one-dimensional oscillators along x y

and z

explicitly we can write them as

proportional to the action of the

raising operators on the ground state

and this here again shows the same

expression for the eigen states but

using the simpler notation that we

typically encounter when working with

tensor product states

finally we've also found that the wave

function of the three-dimensional

quantum harmonic oscillator is simply

given by the product of the wave

functions of three one-dimensional

harmonic oscillators along x along y and

along z

so overall the form of the hamiltonian

of the three-dimensional quantum

harmonic oscillator means that we can

find its solutions by simply combining

the known solutions of various

one-dimensional quantum harmonic

oscillators

we've just seen how the

three-dimensional quantum harmonic

oscillator is a relatively

straightforward extension of its

one-dimensional counterpart the energy

eigenvalues are simply the sum of the

eigenvalues of the one-dimensional

oscillators along the cartesian

directions and the energy eigenstates

are simply the tensor products of the

energy eigenstates of one-dimensional

oscillators along the cartesian

directions this may appear really simple

but actually it provides us with a

really powerful foundation from which to

study a range of really interesting

properties that emerge when we have an

isotropic quantum harmonic oscillator i

therefore encourage you to check out the

video on the degeneracies of the

three-dimensional quantum harmonic

oscillator or the videos where we look

at the isotropic three-dimensional

oscillator at the central potential and

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