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.