Any kind of measurement is organized in space & time. Normally physical quantities are either function of space or function of time and sometimes function of both. In non-relativistic quantum mechanics, though wave function depends on both space and time, only space enters quantum mechanics as operator, time does not. There is no time operator in quantum mechanics. All quantum mechanics textbooks introduce only space as operator, not time. But it is not only possible to produce time operator in quantum mechanics but also we can find out Eigen value of time operator, determines its square’s commutation relation with energy, and prove that it is Hermitian operator; the expectation value of time operator is real for a wave packet and any two wave functions and are Fourier transform of each other, where represents respectively position, time and energy of any quantum mechanical particle.
2. Origin of Time Operator
If wave nature of some quantum mechanical free particle is described by de Broglie wave function 
where, is wave number, is angular frequency.
[Momentum , ]
This is time operator. Though in Zhi-Yong Wang’s paper  there was a minus sign in front of “ ”. We can omit that.
3. Eigen Value of Time Operator
Now we will see how to find out its Eigen value. We have known,
This is the Eigen value of time operator.
We know that the commutation relation between time operator and energy operator is “ ” .
4. Commutation Relation between Square of Time Operator and Energy
Now we will determine the commutation relation between square of time operator and energy.
This is the commutation relation between square of time operator and energy and it is zero. This means, square of time operator and energy commutes.
5. Wave Functions Which Depends on Position and Time Is Fourier Transform of Wave Function Dependent on Position and Energy
We can show that two wave functions and are Fourier transform of each other. If wave function is given by that,
By multiplying Equations (4) and (5),
In the left hand side, We interchange t & t' and get
Substituting the value of A into Equation (3),
We can generalize (8) & (9) respectively and write these equations,
We can rewrite Equation (3) into this form,
Like Equation (12) we can write energy dependent wave function like this,
From Equations (13) & (14) we can see that wave function.
and are each other’s Fourier transform. Until now, wave functions only depend on position, time, and momentum. For the first time, I have showed that wave also depend on energy.
6. Expectation Value of Time Operator
Now we will see that the expectation value of time operator is real.
This is possible only when time operator’s expectation value is real number. So is real. In other papers  , it has proved that time operator is a Hermitian operator. If any operator is Hermitian then its expected value is real. I don’t prove that time operator’s self-adjointness but directly prove that time operator’s expectation value is real.
Definition of time operator and its commutation relation with energy are given in many papers  , but we have shown that its square commutes with energy; and are each other’s Fourier transform and expectation value of time operator is real. We also have determined its Eigen value. Readers of this paper can use time operator and create a new kind of equation of motion which will become alternate of Schrodinger equation.