From ket vectors to wavefunction 2

In our previous post Position Operator: From Ket vectors to wavefunctions we have seen that wavefunctions are the scalar product of ket vectors with position eigenkets. More precisely we call these position space wavefunctions. We can also have momentum space wavefunctions which are the scalar product of the ket vectors with momentum operator eigenkets. We will look at these in some other post. Today we wish to see how do we write different operators and equations in position space.

Most of the introductory books on quantum mechanics introduce the theory in the position space making use of wavefunctions.

Consider the eigenvalue equation for the Hamiltonian operator

\hat H \left|\psi\right\rangle=E\left|\psi\right\rangle

Here for the time being we are not using the subscript on energy E and the eigenket \left|\psi\right\rangle. Now we take the scalar product of the above equation with position eigenket \left|x\right\rangle as

\left\langle x |\hat H|\psi\right\rangle=E\left\langle x|\psi\right\rangle

The right hand side we recognize as the wavefunction

\left\langle x|\psi\right\rangle=\psi(x)

The left side \left\langle x |\hat H|\psi\right\rangle can be evaluated using the following rules

\left\langle x|\hat x|\psi\right\rangle= x \left\langle x|\psi\right\rangle= x \psi(x)

\left\langle x|\hat p|\psi\right\rangle= -i\hbar \frac{\partial}{\partial x} \left\langle x|\psi\right\rangle= -i\hbar \frac{\partial}{\partial x}\psi(x)

The first rule tells us that the position operator in position space is multiplicative and the momentum operator in position space is differential. Using these rules and remembering that the Hamiltonian is the sum of kinetic and potential energy operators

\hat H=\frac {\hat p^2}{2m}+ \hat V(\hat x)

we finally write the eigenvalue equation for the Hamiltonian operator in position space as

\left(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)\right)\psi(x)=E\psi(x).

This is Schrodinger time independent equation. See if you can do the mathematics to write this equation.

 

Exercise Problem

A particle in a box of length L is in the state \left|\psi\right\rangle such that its wavefunction is

\psi(x)=\left\langle x|\psi\right\rangle= \sqrt{\frac{2}{L}}\sin\left(\frac{\pi x}{L} \right)

What is the probability that the particle is in right half of the box?

Solution

The probability density associated with the given wave function is

\mathrm{p}(x) = |\psi(x)|^2= \frac 2 L \sin^2\left(\frac{\pi x}{L}\right)

The probability that the particle is in right half of the box is the integration of this probability density from x=L/2 to x=L. Thus the required probability is

\mathrm{P}= \int_{L/2}^L \mathrm{p}(x) dx = \frac 2 L\int_{L/2}^L\sin^2\left(\frac{\pi x}{L}\right)dx

The result of this integration should be 1/2 (Work out this integration).

Thus the probability that the particle is in right half of the box of length L in the state given in the problem is L/2.

Position Operator: From Ket vectors to wavefunctions

Suppose we are interested in measuring the position of a quantum particle. The observable that will do the job is the position operator. The eigenvalue equation for position operator is

\hat x \left|x\right\rangle=x \left|x\right\rangle

Here \left|x\right\rangle are position eigenkets and x are the corresponding eigenvalues. This operator is different from the one we have already encountered for the energy and will need slightly different treatment. The eigenvalues of this operator are continuous. The completeness relation thus changes into an integration instead of a summation.

Completeness relation

We write the completeness relation for the eigenkets of position operator as

\int_{-\infty}^{\infty}dx \left|x\right\rangle\left\langle x\right|=\hat I

We can use this completeness relation to write any arbitrary ket as

\left|\psi\right\rangle=\hat I \left|\psi\right\rangle=\int_{-\infty}^{\infty}dx \left|x\right\rangle\left\langle x|\psi\right\rangle

Wavefunctions and probability density

The scalar product/expansion coefficient/probablity amplitude  \left\langle x|\psi\right\rangle is called (position space) wavefunction.  We write it as

\psi(x)=\left\langle x|\psi\right\rangle

Wavefunctions are the scalar product of ket vectors with position eigenkets.

The modulus square of the wave function is again related to the probability of the measurement outcome of the position meaurement of the particle. However due to the continuous nature of the eigenvalues of this operator, we need to modify the corresponding postulate (Postulate 4 discussed in the class). The postulate that gives the probabilities of measurement outcome of position measurement of the quantum particle can be stated as follows:

The probability of obtaining the position of the particle in a narrow strip of width dx centered at x=x_0 is given by |\psi(x_0)|^2dx

This can be seen in the figure below

ProbabilityDensity

 

The quantity plotted as the function of x is  |\psi(x)|^2 and is called probability density. Obviously the probability of finding a particle in a location is higher if the probability density is high over there. We can now also answer the questions like “What is the probability of finding the particle between x=a and x=b. This probability will be \int_a^b|\psi(x)|^2dx.

Scalar product as an integral

Another use of the completeness relation is to write the scalar product in the form of an integral. Consider the norm of the vector \left|\psi\right\rangle

\left\langle\psi|\psi\right\rangle=\left\langle\psi|\hat I|\psi\right\rangle=\int_{-\infty}^{\infty}\left\langle\psi|x\right\rangle\left\langle x|\psi\right\rangle dx=\int_{-\infty}^{\infty} \psi^*(x)\psi(x) dx=1

More generally the scalar product of two vectors can be written as

\left\langle\phi|\psi\right\rangle=\left\langle\phi|\hat I|\psi\right\rangle=\int_{-\infty}^{\infty}\left\langle\phi|x\right\rangle\left\langle x|\psi\right\rangle dx=\int_{-\infty}^{\infty} \phi^*(x)\psi(x) dx

Orthogonality relation

The orthogonality condition is also modified in case of continuous spectrum. It is given as follows for the position eigenkets

\left\langle x^\prime|x\right\rangle=\delta(x^\prime-x)

The expression on right is called Dirac delta function. Its value is zero if the argument is non-zero. If the argument is zero then the value of the function is undefined. However the integration over this undefined value is unity. We write it

\int \delta(x)=1.

 

 

 

 

Particle in a box 2

In A Particle in a Box- 1 we have seen that we can write the quantum state of a particle in a box in the eigen-basis of energy operator. From now on we will call this the Hamiltonian operator. For a recap the eigenvalue equation for the Hamiltonian for a particle in a box system is

\hat H \left| \phi_n\right\rangle = E_n \left|\phi_n\right\rangle

with

\left\langle \phi_n|\phi_m\right\rangle=\delta_{nm}

and

\sum_{n=1}^{\infty} \left|\phi_n\right\rangle\left\langle\phi_n\right|=\hat I

With this we could write any arbitrary state of the system as

\left|\psi\right\rangle = \sum_{n=1}^{\infty} \left|\phi_n\right\rangle\left\langle\phi_n|\psi\right\rangle

Here \left\langle\phi_n|\psi\right\rangle are the expansion coefficients (or probability amplitudes) whose modulus square  |\left\langle\phi_n|\psi\right\rangle|^2 is the probability that upon measurement the system will return energy E_n associated with the eigenvector \left|\phi_n\right\rangle.

Lets now ask the question

Where is the particle in the box?

To answer this question we need to look first at Position Operator

 

 

 

 

 

 

 

Summary of what we have learned

Here is a short summary of what we have learned.

  1. Quantum mechanics is a theory that allows us to calculate the probabilities of fundamental interactions taking place in the microscopic world of atoms and molecules.
  2. To calculate these probabilities, and to use quantum mechanics we consider the state of the system which is a vector in a state space associated with the system. We write this vector as a ket (or bra) vector in Dirac bra-ket notation.
  3. The expansion coefficients appearing in the expansion of the state (ket)  vector in some basis are the probability amplitudes whose modulus square is the probability that the system on measurement will result in a value associated with the basis vector. More technically these expansion coefficients are the scalar or inner product of system ket with the basis vectors.
  4. The basis vectors are generally the eigenvectors of an observable (which is a Hermitian operator) associated with the dynamical variable we wish to measure.

 

After this short summary you can look again at A Particle in a Box- 1or to a brief summary in Particle in a box 2.

A Particle in a Box- 1

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In the previous example we have seen that quantum mechanics allows us to calculate the experimentally verifiable probabilities. In this post we will further build our theory by taking another simple example, and by considering the quantum mechanical description of that system.

Let’s consider a particle trapped in a box.

Its classical description comprises the position and momentum information of the particle at some instant of time. From this initial data and using the classical equations of motion we can calculate the future behavior of the particle for all times.

Now lets look at the quantum description of the system.

How to write the state vector for the particle.

From our previous discussion we know that quantum mechanics associates a vector with the state of the particle. So our first task is to work out how to write this vector. For this we consider some experimentally measurable observable — for example the energy of the particle.

Lets assume that the various energy measurements of the particle result in values E_1, E_2,… We can associate a vector with each of these energy values. Now, since the count of such vectors will be infinite ( for infinite possible values of the energy), we can not easily represent these vectors as column matrices as we did in our previous example. So we need to introduce a new compact notation.

A new notation

We adopt a new notation as follows. To each energy value E_n, we associate a vector denoted as \left|\phi_n\right\rangle, with n=1,2,3,...

The vectors |\phi_n\rangle span a space called the state space of the particle, and all possible states of the particle are some linear combination of the vectors \left|\phi_n\right\rangle. We will come to this point in a short time. Lets first consider some properties of the vectors \left|\phi_n\right\rangle.

Scalar product

These vectors \phi_n\rangle satisfy the following relation.
\langle \phi_n|\phi_m\rangle=0

The left side of the above equation represents the scalar ( or inner) product of the two vector \left|\phi_n\right\rangle and \left|\phi_m\right\rangle. The value 0 on the right side tells us that the two vectors are orthogonal to each other. This means that the two vectors are linearly independent, and that the one can not be written in terms of the other.

Now if we consider the scalar product of \left|\phi_n\right\rangle with itself, we get
\langle \phi_n|\phi_n\rangle=1

In this case the scalar product on left hand side is a measure of the norm of the vector (It is the square of the norm of the vector |\phi_n\rangle). From above equation we see that the vectors |\phi_n\rangle are normalized.

The last two relations can be written in a more compact form as
\langle \phi_n|\phi_n\rangle=\delta_{nm}
where \delta_{nm} is called Kronecker delta. Its value is 1 if the two indices in the subscript are the same; else the value is 0.

More generally, the scalar product of two arbitrary vectors \langle \psi|\chi\rangle is a complex number. It quantifies how much the vector |\psi\rangle resembles |\chi\rangle, and it satisfies the following properties.

\langle\psi|\chi\rangle = \langle\chi|\psi\rangle^*

\langle\psi| (a|\chi_1\rangle + b |\chi_2\rangle) =a \langle\psi|\chi_1\rangle + b \langle \psi|\chi_2\rangle

Here a and b are complex numbers.

Outer product

The outer product of the vectors |\phi_n\rangle is an operator, that can transform a vector into another. For example consider

(|\phi_n\rangle\langle\phi_n|)|\psi\rangle = \langle \phi_n|\psi\rangle |\phi_n\rangle

The operator |\phi_n\rangle\langle\phi_n| in this case is a projector that projects any arbitrary vector |\psi\rangle to the vector |\phi_n\rangle. One interesting property of the vectors |\phi_n\rangle is that the sum of all the projectors they give rise to equals identity operator. Mathematically it can be written as
\sum_{n=1}^\infty |\phi_n\rangle\langle\phi_n| = \hat I

This is called the completion relation. It allows us to treat the vectors |\phi_n\rangle as base vectors, and use these base vectors to write any vector, operator, or a scalar product. As example

|\psi\rangle= \hat I |\psi\rangle = \sum_{n=1}^\infty |\phi_n\rangle\langle\phi_n|\psi\rangle = \sum_{n=1}^\infty c_n |\phi_n\rangle

Here, c_n=\langle\phi_n|\psi\rangle are called expansion coefficients for the expansion of |\psi\rangle in the base vectors |\phi_n\rangle.

Finally the vectors |\phi_n\rangle satisfy the equation
\hat H |\phi_n\rangle = E_n |\phi_n\rangle

In this case we see that the vector is not changed after the action of the operator \hat H, and is merely multiplied by a number E_n. This is called an eigenvalue equation. The operator on the left hand side is called the Hamiltonian. It represents the total energy of the system. The above equation tells us that our base vectors |\phi_n\rangle are the eigenvectors of the Hamiltonian, and the eigenvalues associated with these vectors are the experimentally observable energies of the system (E_1, E_2, ... from where we started our discussion).

Lets go back to our particle in a box, and to our question of writing the state vector for the particle. The state of the particle in the box at some time t=0 is given by |\psi(0)\rangle. We can expand it in the eigenstates of the Hamiltonian as

|\psi(0)\rangle = \sum_{n=1}^\infty c_n |\phi_n\rangle
where c_n=\langle\phi_n|\psi(0)\rangle is the overlap or the component of |\psi(0)\rangle along the base vector |\phi_n\rangle.

The time evolution of the system is given by the famous Schrodinger equation

i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle =\hat H |\psi(t)\rangle
Where \hat H is again the Hamiltonian of the system representing the total energy of the system.
The solution of this equation (in our case) is
|\psi(t)\rangle = e^{-i \hat H t/\hbar } |\psi(0)\rangle
Using the eigenvalue equation for the Hamiltonian, and the expansion of the state vector in the eigenvectors of the Hamiltonian we can write the solution as

|\psi(t)\rangle = \sum_{n=1}^\infty c_n e^{-iE_n t/\hbar}|\phi_n\rangle

This is the complete description of the particle in the box for all time.
Lets now see what information we can extract from this description. First lets ask the question what is the energy of the particle. For this we will have to perform an experiment where we measure this energy. If we perform such experiment and we keep in mind that quantum mechanics allows us to calculate the probability of random events happening in the nature, we get the following answer

  • The result of the energy measurement will be a random value among all the eigenvlaue of the  Hamiltonian.
  • The probability of obtaining a given eigenvalue  (say E_n ) is given by |c_n|^2=|\langle\phi_n|\psi(t)\rangle|^2. This is the modulus square of the amplitude associated with the path |\phi_n\rangle.

In our next post we will see how to extract more information from the state vector of the particle.

Nature speaks linear algebra

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In our previous post we considered a simple interferometric  experiment and claimed that quantum mechanics can account for the experimentally observed detection probabilities in the interferometer. In this post we will see how it is done. The idea was to to attach complex amplitudes to all paths available to the photon. Mathematically it can be done by borrowing tools from linear algebra, and describing  the state of the photon in terms of vectors— which can be conveniently represented as matrices. This brings us to the very first lesson.

The states of a quantum system are described by vectors

Let’s again consider the simple Mach-Zehnder interferometer.

img_0391-1

We can identify two paths between the source S, and the detectors D1 and D2. Let’s represent these as follows

  • The  matrix \begin{pmatrix} 1 \\  0\end{pmatrix} represents the state in which the photon is on the left side of the two beam-splitters. We can note that initially when the photon is emitted from the source, it is in state \begin{pmatrix} 1 \\ 0\end{pmatrix}.
  • The  matrix \begin{pmatrix} 0 \\ 1\end{pmatrix} represents the state in which the photon is on the right side of the beam-splitters.

The photon in the state \begin{pmatrix} 1\\0 \end{pmatrix} approaches the first beam-splitter. The beam-splitter splits the state of the photon in two parts. Please note here  that the splitting takes place for the state which is a vector, and not for the photon which is a physical entity.

The splitting takes place for the state which is a vector, and not for the photon which is a physical entity.

Mathematically the beam-splitter action can be represented as follows

\begin{pmatrix} ir & t \\ t & ir \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} =ir \begin{pmatrix} 1 \\ 0 \end{pmatrix} +t \begin{pmatrix} 0 \\ 1\end{pmatrix}

The first 2×2 matrix in the above equation represents the action of the beam-splitter.

The next coulumn matrix is the initial state of the photon. The right side of the equation is the state of the photon after passing through the first beam-splitter.

Here we see that the moudulus square of the complex amplitudes assoicated with the two paths give us the correct reflection and transmission probabiliteies r^2 and t^2, respectively.

Also note that i =\sqrt{-1} is the phase difference between the reflected and transmitted amplitudes. It comes from the requirement of unitarity of beam-splitter transformation— a concept that we will learn in another post.

After the first beam-splitter we have two mirrors which do not change the state of the photon. Finally, the photon approaches the second beam-splitter in the state \begin{pmatrix} ir \\ t \end {pmatrix}. The action of the second beam-splitter is given by

\begin{pmatrix} ir & t \\ t & ir \end{pmatrix} \begin{pmatrix} ir \ \ t \end{pmatrix} =(-r^2+t^2 )\begin{pmatrix} 1 \\ 0 \end{pmatrix} +2 irt \begin{pmatrix} 0\\ 1 \end{pmatrix}

The right hand side of the above equation yields the state of the photon after the second beam-splitter. The modulus square of the amplitudes
(t^2-r^2)^2 and  4r^2t^2 correctly give the experimentally verifiable detection probabailities of D1 and D2 detectors, respectively.

Summary

Here is a quick review of what we have learned so far

  • The state of a quantum system is described by a vector which can be represented as  a coulumn matrix. We will later see that we can also represent the states as row matrices, or using analytic  functions.
  • The operations on quantum systems are carried out by operators that can be represented as square matrices.
  • The modulus  square of the complex amplitudes are related to experimentally observed probabilities.

What is quantum mechanics?

A photon approached a semi-transparent mirror and asked “Will you let me pass through?”. The mirror kept silent for a minute and then replied “I don’t know. The god has not yet rolled the dice”.

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Quantum mechanics is a theory that allows us to calculate the probabilities of random events happening in the nature at the very fundamental level. The emphasis here is on the words random and the probabilities. The word random tells us that the most basic events in the nature, like excitation of an atom or the reflection of a photon by a mirror, can not be predicted. In fact nature exhibits true randomness only in the quantum world. All other apparently random phenomenon is actually pseudo-random — governed by some complicated equation or a fancy algorithm. One exception to the last statement might be the randomness of our thought processes. But it is much safer not to ponder on this question.

So, if we can’t predict any fundamental event with certainty, what we do? We calculate the probabilities of the events we are interested in. This is what nature allows us to do, and quantum mechanics teaches us how to do.

Let’s take an example. Consider a photon incident on a beam-splitter . A beam-splitter is a device that splits a beam of light into a reflected part and a transmitted part. We may ask the question whether the photon incident on the beam-splitter will be reflected back or will be transmitted through. What comes next will amaze many. The answer to this question is that we don’t know. The nature does not have the answer to this stupidly simple question (or if it has, it has not yet told us).

In a sense it is almost sacred and frightening. On one hand nature has allowed us extraordinary scientific and technological feasts. It has allowed us to decode and play with the human genome. It has let us create new life forms. It has allowed us to hear the whispers of the merging of the black holes happened in a distant past. And it has brought us to the verge of the next big step in evolution in which we might be replaced by the machines we create. Yet it does not let us answer the simple question whether the photon will pass through the beam-splitter.

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Coming back to our problem, if we run the experiment — striking a photon with the beam-splitter— a large number of times, we will find that the beam-splitter reflects the photon with the probability r^2 , and transmits it with the probability t^2 with r^2+t^2 =1 . This is all we can tell about this simple experimental setup.

Let’s consider another experiment . In this one we introduce two mirrors and a second identical beam-splitter to the setup.

img_0391-1

For this setup our intuition tells us that since both beam-splitters reflect and transmit the photon with the probabilities r^2 and t^2 , respectively, the probability that the photon is detected at the detector D1 should be r^4 +  t^4. And the probability that it is detected at D2 should be  2r^2 t^2 . However, this is not the case. If we perform the experiment a large number of times, we will see that a (t^2-r^2)^2 fraction of all incident photons is detected at the detector D1, and 4r^2t^2 fraction goes to the detector D2. We see here that for r=t , the detector D1 will never click. Moreover, we can see that by changing the relative path length along the two arms between the beam-splitters, we can change the probabilities of detection at the two detectors all the way from zero to one. This seems very much like an interference phenomenon in which by changing the path difference between the interfering paths we can get either the constructive or the destructive interference. Indeed it is the case in our present example.

In order to account for the experimental observations, and to understand the interference phenomenon discussed above we add this rule to our theory. If a quantum system can follow different paths, it will follow all those paths. To each path we can associate a complex amplitude which tells us how likely the quantum system is to follow that path. When different path reunite we add the corresponding amplitudes. Finally at the time of the measurement (for example the detection at detector D1), the modulus square of the complex amplitude gives the probability of detection. This is how quantum mechanics accounts for how nature behaves at the very fundamental level.

The setup shown in the last example is called the Mach-Zehnder interferometer. It holds some very deep mysteries of the nature. In the future posts we will try to appreciate some of these mysteries.

Hello World

Hello world! I am Faheel Hashmi and I wish to talk about quantum mechanics. This is a theory that describes the nature at its most basic level, and the nature at this level is very different from how we experience it in our everyday life.  I have taught quantum mechanics a couple of times, and in this blog I wish to share with you the excitement of learning this wonderful theory.

The Other Stuff portion of the blog will occasionally have posts on education system, science in general, some issues we face in our third world society, the never ending debate on mind and matter.

In time this site might transform into my personal webpage containing all my lecture notes, my research, presentations and articles related to my interests, and maybe a forum to interact with my students.

In the end I wish to introduce my crew. This includes an artist from the future who will be rendering most of the artwork for this blog. Meet my daughter Urwa. She is in grade 2, and has most generously  allowed me to use some of her artwork for my blog, and has promised to provide me the new material on regular basis. My younger daughter Ayesha will contribute to the blog by letting me do my work in peace. Lastly my wonderful wife Sidra is the main motivation and the source of encouragement for this project. She will also be proofreading my posts. So if you find typo and grammatical errors in my posts, you know who to blame.

Thanks for visiting my site. I hope you will visit it often, and will find the content on this site worth your time.