Chapter 10, Problem 10.1E

State the postulates of quantum mechanics introduced throughout the chapter in your own words.

Interpretation Introduction

Interpretation:

The postulates of quantum mechanics are to be stated.

Concept introduction:

In quantum mechanics, the wavefunction is given by $\Psi$. The wavefunction contains all the information about the state of a system. The wavefunction is the function of the coordinates of particles and time.

Answer to Problem 10.1E

Postulate $\text{I}$ states that the wavefunction $\Psi$ contains all the information about the state of the system.

Postulate $\text{II}$ states that to determine the value of an observable, some mathematical operation called operator is to be performed on a wavefunction.

Postulate $\text{III}$ states that the eigen values of the wavefunction that are obtained when an operator is applied are the only possible values of the observables.

Postulate $\text{IV}$ states that the wavefunctions must satisfy the time dependent Schrodinger equation.

Postulate $\text{V}$ states that for the normalized wavefunctions, the average value or the expectation value of an observable $\langle \text{O}\rangle$ given by,

$\langle \text{O}\rangle ={\displaystyle \underset{\begin{array}{l}\text{all}\\ \text{space}\end{array}}{\int }{\Psi }^{\ast }\widehat{\text{O}}\Psi d\tau }$

Postulate $\text{VI}$ states that for any quantum mechanical operator, the set of eigen functions are representing the complete mathematical set of functions.

Postulate $\text{VII}$ states if for a given system, the linear combination of non -degenerate wavefunctions ${\Psi }_n$ given by wavefunction $\Psi$ having eigen values $a_n$ is,

$\Psi ={\displaystyle \sum c_n{\Psi }_n}$

The eigen value equation for wavefunctions ${\Psi }_n$ is given by,

$\widehat{A}{\Psi }_n=a_n{\Psi }_n$

Explanation of Solution

The postulates of quantum mechanics are,

Postulate $\text{I}$ states that the wavefunction $\Psi$ contains all the information about the state of the system. The wavefunction is the function of the coordinates of particles and time. The wavefunction must be single-valued, continuous, differentiable and bounded.

Postulate $\text{II}$ states that to determine the value of an observable, some mathematical operation is to be performed on a wavefunction. This mathematical operation is known as operator. Operator acts on a function in order to produce a function. For every physical observable there exists a corresponding operator.

Postulate $\text{III}$ states that the eigen values of the wavefunction that are obtained when an operator is applied are the only possible values of observables. The expression for the eigen value is given by,

$\widehat{A}\Psi =a\Psi$

where, $\widehat{A}$ is the operator that operates on wavefunction $\Psi$ and $a$ is the eigen value.

Postulate $\text{IV}$ states that the wavefunctions must satisfy the time dependent Schrodinger equation. The time dependent Schrodinger equation is,

$\widehat{H}\Psi =i\hslash \frac{\partial \Psi }{\partial t}$

Postulate $\text{V}$ states that for the normalized wavefunctions the average value or the expectation value of an observable $\langle \text{O}\rangle$ whose operator is $\widehat{\text{O}}$ is given by the expression,

$\langle \text{O}\rangle ={\displaystyle \underset{\begin{array}{l}\text{all}\\ \text{space}\end{array}}{\int }{\Psi }^{\ast }\widehat{\text{O}}\Psi d\tau }$

Where, the operator $\widehat{\text{O}}$ operates on the wavefunction $\Psi$.

Postulate $\text{VI}$ states that for any quantum mechanical operator, the set of eigen functions are representing the complete mathematical set of functions.

Postulate $\text{VII}$ states if for a given system, the linear combination of non-degenerate wavefunctions ${\Psi }_n$ given by wavefunction $\Psi$ having eigen values $a_n$ is,

$\Psi ={\displaystyle \sum c_n{\Psi }_n}$

The eigen value equation for wavefunctions ${\Psi }_n$ is given by,

$\widehat{A}{\Psi }_n=a_n{\Psi }_n$

The probability that $a_n$ is the value of the given measurement is ${\left|c_n\right|}^2$. The combination of all possible ${\Psi }_n$’s to give $\Psi$ is known as superposition principle.

Conclusion

Postulate $\text{I}$ states that the wavefunction $\Psi$ contains all the information about the state of the system.

Postulate $\text{II}$ states that to determine the value of an observable, some mathematical operation is to be performed on a wavefunction. This mathematical operation is known as operator.

Postulate $\text{III}$ states that the eigen values of the wavefunction that are obtained when an operator is applied are the only possible values of observables.

Postulate $\text{IV}$ states that the wavefunctions must satisfy the time dependent Schrodinger equation.

Postulate $\text{V}$ For the normalized wavefunctions the average value or the expectation value of an observable $\langle \text{O}\rangle$ given by,

$\langle \text{O}\rangle ={\displaystyle \underset{\begin{array}{l}\text{all}\\ \text{space}\end{array}}{\int }{\Psi }^{\ast }\widehat{\text{O}}\Psi d\tau }$

Postulate $\text{VI}$ states that for any quantum mechanical operator, the set of eigen functions are representing the complete mathematical set of functions.

Postulate $\text{VII}$ states if for a given system, the linear combination of non-degenerate wavefunctions ${\Psi }_n$ given by wavefunction $\Psi$ having eigen values $a_n$ is,

$\Psi ={\displaystyle \sum c_n{\Psi }_n}$

The eigen value equation for wavefunctions ${\Psi }_n$ is given by,

$\widehat{A}{\Psi }_n=a_n{\Psi }_n$