1. Quantum One

3. Postulate IV

4. The Evolution of Quantum Mechanical Systems
Postulate IV
The Evolution of Quantum Mechanical Systems

5. We are now ready to finish up the set of postulates that we have been developing to describe the formalism of quantum mechanics.
The last postulate describes the way a quantum mechanical system behaves in between the times during which measurements are being made.
As we have seen, during a measurement process, a quantum mechanical system, in contact with a classical measuring device, evolves non-deterministically as the state vector collapses into one of the eigenspaces of the particular observable being measured.
In between these measurement events, evolution of the state is governed by the fourth postulate.

6. We are now ready to finish up the set of postulates that we have been developing to describe the formalism of quantum mechanics.
The last postulate describes the way a quantum mechanical system behaves in between the times during which measurements are being made.
As we have seen, during a measurement process, a quantum mechanical system, in contact with a classical measuring device, evolves non-deterministically as the state vector collapses into one of the eigenspaces of the particular observable being measured.
In between these measurement events, evolution of the state is governed by the fourth postulate.

7. We are now ready to finish up the set of postulates that we have been developing to describe the formalism of quantum mechanics.
The last postulate describes the way a quantum mechanical system behaves in between the times during which measurements are being made.
As we have seen, during a measurement process, a quantum mechanical system, in contact with a classical measuring device, evolves non-deterministically as the state vector collapses into one of the eigenspaces of the particular observable being measured.
In between these measurement events, evolution of the state is governed by the fourth postulate.

8. We are now ready to finish up the set of postulates that we have been developing to describe the formalism of quantum mechanics.
The last postulate describes the way a quantum mechanical system behaves in between the times during which measurements are being made.
As we have seen, during a measurement process, a quantum mechanical system, in contact with a classical measuring device, evolves non-deterministically as the state vector collapses into one of the eigenspaces of the particular observable being measured.
In between these measurement events, evolution of the state is governed by the fourth postulate.

9. Postulate IV (Evolution)
Between measurements, the state vector |𝜓(𝑡)〉 of a quantum mechanical system evolves deterministically according to Schrödinger's equation of motion
in which 𝐻=𝐻(𝑡), referred to as the Hamiltonian operator, is the observable associated with the total energy of the system at time 𝑡.

10. Postulate IV (Evolution)
Between measurements, the state vector |𝜓(𝑡)〉 of a quantum mechanical system evolves deterministically according to Schrödinger's equation of motion
in which 𝐻=𝐻(𝑡), referred to as the Hamiltonian operator, is the observable associated with the total energy of the system at time 𝑡.
Note that this is a first order, linear differential equation for the state vector |𝜓(𝑡)〉.

11. Postulate IV (Evolution)
Between measurements, the state vector |𝜓(𝑡)〉 of a quantum mechanical system evolves deterministically according to Schrödinger's equation of motion
in which 𝐻=𝐻(𝑡), referred to as the Hamiltonian operator, is the observable associated with the total energy of the system at time 𝑡.
Note that this is a first order, linear differential equation for the state vector |𝜓(𝑡)〉.

12. In practice, to use Schrödinger's equation we project it onto the basis vectors of an appropriate representation.
Thus, if the vectors {|𝑛〉} form an ONB for the space of interest, then we can write
Sliding 〈𝑛| past the time derivative and inserting a complete set of states|𝑛′〉 to the right of the Hamiltonian, we obtain
in which we recognize coefficients for the time dependent expansion

13. In practice, to use Schrödinger's equation we project it onto the basis vectors of an appropriate representation.
Thus, if the vectors {|𝑛〉} form an ONB for the space of interest,
Sliding 〈𝑛| past the time derivative and inserting a complete set of states|𝑛′〉 to the right of the Hamiltonian, we obtain
in which we recognize coefficients for the time dependent expansion

14. In practice, to use Schrödinger's equation we project it onto the basis vectors of an appropriate representation.
Thus, if the vectors {|𝑛〉} form an ONB for the space of interest, then we can multiply on the left by
Sliding 〈𝑛| past the time derivative and inserting a complete set of states |𝑛′〉 to the right of the Hamiltonian, we obtain
in which we recognize coefficients for the time dependent expansion

15. In practice, to use Schrödinger's equation we project it onto the basis vectors of an appropriate representation.
Thus, if the vectors {|𝑛〉} form an ONB for the space of interest, then we can multiply on the left by
Sliding 〈𝑛| past the time derivative and inserting a complete set of states |𝑛′〉 to the right of the Hamiltonian, we obtain
in which we recognize coefficients for the time dependent expansion

16. In practice, to use Schrödinger's equation we project it onto the basis vectors of an appropriate representation.
Thus, if the vectors {|𝑛〉} form an ONB for the space of interest, then we can multiply on the left by
Sliding 〈𝑛| past the time derivative and inserting a complete set of states |𝑛′〉 to the right of the Hamiltonian, we obtain
in which we recognize coefficients for the time dependent expansion

17. Thus, in this representation, Schrödinger's equation takes the form
of a set of first-order, generally coupled differential equations for the time- dependent expansion coefficients for the state |𝜓(𝑡)〉 in this basis.

18. In a continuous representation |𝛼〉, the state of the system is represented by the wave function 𝜓(𝛼,𝑡) =〈𝛼|𝜓(𝑡)〉 and the Hamiltonian becomes, in general, an integro-differential operator acting on this function.
In the most general case, the matrix elements of 𝐻 between the continuous basis states |𝛼〉 are defined by some kernel 𝐻(𝛼,𝛼′)=〈𝛼|𝐻|𝛼′〉.
Projection of the Schrödinger equation onto the basis states of such a representation then leads to the expression

19. In a continuous representation |𝛼〉, the state of the system is represented by the wave function 𝜓(𝛼,𝑡) =〈𝛼|𝜓(𝑡)〉 and the Hamiltonian becomes, in general, an integro-differential operator acting on this function.
In the most general case, the matrix elements of 𝐻 between the continuous basis states |𝛼〉 are defined by some kernel 𝐻(𝛼,𝛼′)=〈𝛼|𝐻|𝛼′〉.
Projection of the Schrödinger equation onto the basis states of such a representation then leads to the expression

20. In a continuous representation |𝛼〉, the state of the system is represented by the wave function 𝜓(𝛼,𝑡) =〈𝛼|𝜓(𝑡)〉 and the Hamiltonian becomes, in general, an integro-differential operator acting on this function.
In the most general case, the matrix elements of 𝐻 between the continuous basis states |𝛼〉 are defined by some kernel 𝐻(𝛼,𝛼′)=〈𝛼|𝐻|𝛼′〉.
Projection of the Schrödinger equation onto the basis states of such a representation then leads to the expression

21. In a continuous representation |𝛼〉, the state of the system is represented by the wave function 𝜓(𝛼,𝑡) =〈𝛼|𝜓(𝑡)〉 and the Hamiltonian becomes, in general, an integro-differential operator acting on this function.
In the most general case, the matrix elements of 𝐻 between the continuous basis states |𝛼〉 are defined by some kernel 𝐻(𝛼,𝛼′)=〈𝛼|𝐻|𝛼′〉.
Projection of the Schrödinger equation onto the basis states of such a representation then leads to the expression

22. As before, taking the derivative with respect to time is a linear operation, so we can write
where the exact derivative of the vector |𝜓(𝑡)〉 (which only depends on 𝑡, not 𝛼) turns into a partial derivative when it acts on the wave function
𝜓(𝛼,𝑡)=〈𝛼|𝜓(𝑡)〉,
which is a function of two variables. Making this substitution and inserting a complete set of states between 𝐻 and |𝜓(𝑡)〉 on the right we obtain an integral equation
for the wave function 𝜓(𝛼,𝑡).

23. As before, taking the derivative with respect to time is a linear operation, so we can write
where the exact derivative of the vector |𝜓(𝑡)〉 (which only depends on 𝑡, not 𝛼) turns into a partial derivative when it acts on the wave function
𝜓(𝛼,𝑡)=〈𝛼|𝜓(𝑡)〉,
which is a function of two variables. Making this substitution and inserting a complete set of states between 𝐻 and |𝜓(𝑡)〉 on the right we obtain an integral equation
for the wave function 𝜓(𝛼,𝑡).

24. As before, taking the derivative with respect to time is a linear operation, so we can write
where the exact derivative of the vector |𝜓(𝑡)〉 (which only depends on 𝑡, not 𝛼) turns into a partial derivative when it acts on the wave function
𝜓(𝛼,𝑡)=〈𝛼|𝜓(𝑡)〉,
which is a function of two variables. Making this substitution and inserting a complete set of states between 𝐻 and |𝜓(𝑡)〉 on the right we obtain an integral equation
for the wave function 𝜓(𝛼,𝑡).

25. As before, taking the derivative with respect to time is a linear operation, so we can write
where the exact derivative of the vector |𝜓(𝑡)〉 (which only depends on 𝑡, not 𝛼) turns into a partial derivative when it acts on the wave function
𝜓(𝛼,𝑡)=〈𝛼|𝜓(𝑡)〉,
which is a function of two variables. Making this substitution and inserting a complete set of states between 𝐻 and |𝜓(𝑡)〉 on the right we obtain an integral equation
for the wave function 𝜓(𝛼,𝑡).

26. Under certain special situations (which occur rather often) the matrix elements of H will involve derivatives of delta functions, and the integral equation will reduce to a differential equation, as we have seen occur with the energy eigenvalue equation in the position representation.
Thus, for a single particle in 3D moving under the influence of a potential, the Hamiltonian is simply the sum of the kinetic and potential energy operators
Under these circumstances, the Schrödinger equation can be written
Projecting on to the basis vectors of the position representation, …

27. Under certain special situations (which occur rather often) the matrix elements of H will involve derivatives of delta functions, and the integral equation will reduce to a differential equation, as we have seen occur with the energy eigenvalue equation in the position representation.
Thus, for a single particle in 3D moving under the influence of a potential, the Hamiltonian is simply the sum of the kinetic and potential energy operators
Under these circumstances, the Schrödinger equation can be written
Projecting on to the basis vectors of the position representation, …

28. Under certain special situations (which occur rather often) the matrix elements of H will involve derivatives of delta functions, and the integral equation will reduce to a differential equation, as we have seen occur with the energy eigenvalue equation in the position representation.
Thus, for a single particle in 3D moving under the influence of a potential, the Hamiltonian is simply the sum of the kinetic and potential energy operators
Under these circumstances, the Schrödinger equation can be written
Projecting on to the basis vectors of the position representation, …

29. Under certain special situations (which occur rather often) the matrix elements of H will involve derivatives of delta functions, and the integral equation will reduce to a differential equation, as we have seen occur with the energy eigenvalue equation in the position representation.
Thus, for a single particle in 3D moving under the influence of a potential, the Hamiltonian is simply the sum of the kinetic and potential energy operators
Under these circumstances, the Schrödinger equation can be written
Projecting on to the basis vectors of the position representation, …

30. we recognize Schrödinger's original equation
Alternatively, we can choose to work in the momentum or wavevector representation, to obtain:

31. in which we recognize Schrödinger's original equation
Alternatively, we can choose to work in the momentum or wavevector representation, to obtain:

32. in which we recognize Schrödinger's original equation
Alternatively, we can choose to work in the momentum or wavevector representation, to obtain:

33. in which we recognize Schrödinger's original equation
Alternatively, we can choose to work in the momentum or wavevector representation, to obtain:

34. Some General Features of Quantum Mechanical Evolution
Determinism - Note that Schrödinger's equation governing the evolution of the state vector is first order in time.
This means that the solution depends only on the initial state of the system, and not, e.g., on its initial rate of change.
Thus, any initial state |𝜓(𝑡₀)〉 of the system at time 𝑡₀ will evolve into a single unique vector |𝜓(𝑡)〉 at some later time 𝑡>𝑡₀.
We note that this implicitly defines a mapping of the space onto itself, and thus implies the existence of an operator 𝑈, or a family of operators 𝑈(𝑡,𝑡₀), that map an arbitrary state at time 𝑡₀ onto the state into which it evolve in time 𝑡.
Thus, we define the evolution operator 𝑈(𝑡,𝑡₀) through the relation
|𝜓(𝑡)〉=𝑈(𝑡,𝑡₀)|𝜓(𝑡₀)〉

35. Some General Features of Quantum Mechanical Evolution
Determinism - Note that Schrödinger's equation governing the evolution of the state vector is first order in time.
This means that the solution depends only on the initial state of the system, and not, e.g., on its initial rate of change.
Thus, any initial state |𝜓(𝑡₀)〉 of the system at time 𝑡₀ will evolve into a single unique vector |𝜓(𝑡)〉 at some later time 𝑡>𝑡₀.
We note that this implicitly defines a mapping of the space onto itself, and thus implies the existence of an operator 𝑈, or a family of operators 𝑈(𝑡,𝑡₀), that map an arbitrary state at time 𝑡₀ onto the state into which it evolve in time 𝑡.
Thus, we define the evolution operator 𝑈(𝑡,𝑡₀) through the relation
|𝜓(𝑡)〉=𝑈(𝑡,𝑡₀)|𝜓(𝑡₀)〉

36. Some General Features of Quantum Mechanical Evolution
Determinism - Note that Schrödinger's equation governing the evolution of the state vector is first order in time.
This means that the solution at time 𝑡 depends only on the initial state of the system, and not, e.g., on its initial rate of change.
Thus, any initial state |𝜓(𝑡₀)〉 of the system at time 𝑡₀ will evolve into a single unique vector |𝜓(𝑡)〉 at some later time 𝑡>𝑡₀.
We note that this implicitly defines a mapping of the space onto itself, and thus implies the existence of an operator 𝑈, or a family of operators 𝑈(𝑡,𝑡₀), that map an arbitrary state at time 𝑡₀ onto the state into which it evolve in time 𝑡.
Thus, we define the evolution operator 𝑈(𝑡,𝑡₀) through the relation
|𝜓(𝑡)〉=𝑈(𝑡,𝑡₀)|𝜓(𝑡₀)〉

37. Some General Features of Quantum Mechanical Evolution
Determinism - Note that Schrödinger's equation governing the evolution of the state vector is first order in time.
This means that the solution at time 𝑡 depends only on the initial state of the system, and not, e.g., on its initial rate of change.
Thus, any initial state |𝜓(𝑡₀)〉 of the system at time 𝑡₀ will evolve into a single unique vector |𝜓(𝑡)〉 at some later time 𝑡>𝑡₀.
We note that this implicitly defines a mapping of the space onto itself, and thus implies the existence of an operator 𝑈, or a family of operators 𝑈(𝑡,𝑡₀), that map an arbitrary state at time 𝑡₀ onto the state into which it evolve in time 𝑡.
Thus, we define the evolution operator 𝑈(𝑡,𝑡₀) through the relation
|𝜓(𝑡)〉=𝑈(𝑡,𝑡₀)|𝜓(𝑡₀)〉

38. Some General Features of Quantum Mechanical Evolution
Determinism - Note that Schrödinger's equation governing the evolution of the state vector is first order in time.
This means that the solution at time 𝑡 depends only on the initial state of the system, and not, e.g., on its initial rate of change.
Thus, any initial state |𝜓(𝑡₀)〉 of the system at time 𝑡₀ will evolve into a single unique vector |𝜓(𝑡)〉 at some later time 𝑡>𝑡₀.
We note that this implicitly defines a mapping of the space onto itself, and thus implies the existence of an operator 𝑈, or a family of operators 𝑈(𝑡,𝑡₀), that map an arbitrary state at time 𝑡₀ onto the state into which it evolves in time 𝑡.
Thus, we define the evolution operator 𝑈(𝑡,𝑡₀) through the relation
|𝜓(𝑡)〉=𝑈(𝑡,𝑡₀)|𝜓(𝑡₀)〉

39. Some General Features of Quantum Mechanical Evolution
Determinism - Note that Schrödinger's equation governing the evolution of the state vector is first order in time.
This means that the solution at time 𝑡 depends only on the initial state of the system, and not, e.g., on its initial rate of change.
Thus, any initial state |𝜓(𝑡₀)〉 of the system at time 𝑡₀ will evolve into a single unique vector |𝜓(𝑡)〉 at some later time 𝑡>𝑡₀.
We note that this implicitly defines a mapping of the space onto itself, and thus implies the existence of an operator 𝑈, or a family of operators 𝑈(𝑡,𝑡₀), that map an arbitrary state at time 𝑡₀ onto the state into which it evolves in time 𝑡.
Thus, we define the evolution operator 𝑈(𝑡,𝑡₀) through the relation
|𝜓(𝑡)〉=𝑈(𝑡,𝑡₀)|𝜓(𝑡₀)〉

40. Some Features of Quantum Mechanical Evolution
Linearity - The linearity of the equations of motion imply a superposition principle for the solutions of the Schrödinger equation.
That is if |𝜓₁(𝑡)〉 and |𝜓₂(𝑡)〉 are two solutions to the Schrödinger equation, that have evolved from initial states |𝜓₁(𝑡₀)〉 and |𝜓₂(𝑡₀)〉,
then the time-dependent vector
|𝜓(𝑡)〉=𝛼|𝜓₁(𝑡)〉+𝛽|𝜓₂(𝑡)〉
is also a solution to the Schrödinger equation for any complex constants 𝛼 and 𝛽, since

41. Some Features of Quantum Mechanical Evolution
Linearity - The linearity of the equations of motion imply a superposition principle for the solutions of the Schrödinger equation.
That is if |𝜓₁(𝑡)〉 and |𝜓₂(𝑡)〉 are two solutions to the Schrödinger equation, that have evolved from initial states |𝜓₁(𝑡₀)〉 and |𝜓₂(𝑡₀)〉,
then the time-dependent vector
|𝜓(𝑡)〉=𝛼|𝜓₁(𝑡)〉+𝛽|𝜓₂(𝑡)〉
is also a solution to the Schrödinger equation for any complex constants 𝛼 and 𝛽, since

42. Some Features of Quantum Mechanical Evolution
Linearity - The linearity of the equations of motion imply a superposition principle for the solutions of the Schrödinger equation.
That is if |𝜓₁(𝑡)〉 and |𝜓₂(𝑡)〉 are two solutions to the Schrödinger equation, that have evolved from initial states |𝜓₁(𝑡₀)〉 and |𝜓₂(𝑡₀)〉,
then the time-dependent vector
|𝜓(𝑡)〉=𝛼|𝜓₁(𝑡)〉+𝛽|𝜓₂(𝑡)〉
is also a solution to the Schrödinger equation for any complex constants 𝛼 and 𝛽, since

43. Some Features of Quantum Mechanical Evolution
Linearity - The linearity of the equations of motion imply a superposition principle for the solutions of the Schrödinger equation.
That is if |𝜓₁(𝑡)〉 and |𝜓₂(𝑡)〉 are two solutions to the Schrödinger equation, that have evolved from initial states |𝜓₁(𝑡₀)〉 and |𝜓₂(𝑡₀)〉,
then the time-dependent vector
|𝜓(𝑡)〉=𝛼|𝜓₁(𝑡)〉+𝛽|𝜓₂(𝑡)〉
is also a solution to the Schrödinger equation for any complex constants 𝛼 and 𝛽, since

44. Some Features of Quantum Mechanical Evolution
Linearity - The linearity of the equations of motion imply a superposition principle for the solutions of the Schrödinger equation.
That is if |𝜓₁(𝑡)〉 and |𝜓₂(𝑡)〉 are two solutions to the Schrödinger equation, that have evolved from initial states |𝜓₁(𝑡₀)〉 and |𝜓₂(𝑡₀)〉,
then the time-dependent vector
|𝜓(𝑡)〉=𝛼|𝜓₁(𝑡)〉+𝛽|𝜓₂(𝑡)〉
is also a solution to the Schrödinger equation for any complex constants 𝛼 and 𝛽, since

45. Some Features of Quantum Mechanical Evolution
Linearity - The linearity of the equations of motion imply a superposition principle for the solutions of the Schrödinger equation.
That is if |𝜓₁(𝑡)〉 and |𝜓₂(𝑡)〉 are two solutions to the Schrödinger equation, that have evolved from initial states |𝜓₁(𝑡₀)〉 and |𝜓₂(𝑡₀)〉,
then the time-dependent vector
|𝜓(𝑡)〉=𝛼|𝜓₁(𝑡)〉+𝛽|𝜓₂(𝑡)〉
is also a solution to the Schrödinger equation for any complex constants 𝛼 and 𝛽, since

46. Some Features of Quantum Mechanical Evolution
Linearity - The linearity of the equations of motion imply a superposition principle for the solutions of the Schrödinger equation.
That is if |𝜓₁(𝑡)〉 and |𝜓₂(𝑡)〉 are two solutions to the Schrödinger equation, that have evolved from initial states |𝜓₁(𝑡₀)〉 and |𝜓₂(𝑡₀)〉,
then the time-dependent vector
|𝜓(𝑡)〉=𝛼|𝜓₁(𝑡)〉+𝛽|𝜓₂(𝑡)〉
is also a solution to the Schrödinger equation for any complex constants 𝛼 and 𝛽, since

47. Some Features of Quantum Mechanical Evolution
Linearity - The linearity of the equations of motion imply a superposition principle for the solutions of the Schrödinger equation.
That is if |𝜓₁(𝑡)〉 and |𝜓₂(𝑡)〉 are two solutions to the Schrödinger equation, that have evolved from initial states |𝜓₁(𝑡₀)〉 and |𝜓₂(𝑡₀)〉,
then the time-dependent vector
|𝜓(𝑡)〉=𝛼|𝜓₁(𝑡)〉+𝛽|𝜓₂(𝑡)〉
is also a solution to the Schrödinger equation for any complex constants 𝛼 and 𝛽, since

48. Some Features of Quantum Mechanical Evolution
Linearity - The linearity of the equations of motion imply a superposition principle for the solutions of the Schrödinger equation.
That is if |𝜓₁(𝑡)〉 and |𝜓₂(𝑡)〉 are two solutions to the Schrödinger equation, that have evolved from initial states |𝜓₁(𝑡₀)〉 and |𝜓₂(𝑡₀)〉,
then the time-dependent vector
|𝜓(𝑡)〉=𝛼|𝜓₁(𝑡)〉+𝛽|𝜓₂(𝑡)〉
is also a solution to the Schrödinger equation for any complex constants 𝛼 and 𝛽, since
In fact, |𝜓(𝑡)〉 is the state that evolved from |𝜓(𝑡₀)〉=𝛼|𝜓₁(𝑡₀)〉+𝛽|𝜓₂(𝑡₀)〉

49. Some Features of Quantum Mechanical Evolution
Linearity - The linearity of the equations of motion imply a superposition principle for the solutions of the Schrödinger equation.
That is if |𝜓₁(𝑡)〉 and |𝜓₂(𝑡)〉 are two solutions to the Schrödinger equation, that have evolved from initial states |𝜓₁(𝑡₀)〉 and |𝜓₂(𝑡₀)〉,
then the time-dependent vector
|𝜓(𝑡)〉=𝛼|𝜓₁(𝑡)〉+𝛽|𝜓₂(𝑡)〉
is also a solution to the Schrödinger equation for any complex constants 𝛼 and 𝛽, since
This also implies that the evolution operator 𝑈(𝑡,𝑡₀) is a linear operator.

50. Some Features of Quantum Mechanical Evolution
Conservation of the Norm - It is also relatively easy to show that quantum mechanical evolution preserves the norm of the state vector, a condition which is obviously important if we wish the total sum of probabilities to be conserved.
Thus, we consider the rate of change of the (squared) length of a vector |𝜓(𝑡)〉 evolving under the Schrödinger equation
where we have simply used the chain rule on the right hand side.
From the Schrödinger equation itself we deduce that
the adjoint of which gives

51. Some Features of Quantum Mechanical Evolution
Conservation of the Norm - It is also relatively easy to show that quantum mechanical evolution preserves the norm of the state vector, a condition which is obviously important if we wish the total sum of probabilities to be conserved.
Thus, we consider the rate of change of the (squared) length of a vector |𝜓(𝑡)〉 evolving under the Schrödinger equation
where we have simply used the chain rule on the right hand side.
From the Schrödinger equation itself we deduce that
the adjoint of which gives

52. Some Features of Quantum Mechanical Evolution
Conservation of the Norm - It is also relatively easy to show that quantum mechanical evolution preserves the norm of the state vector, a condition which is obviously important if we wish the total sum of probabilities to be conserved.
Thus, we consider the rate of change of the (squared) length of a vector |𝜓(𝑡)〉 evolving under the Schrödinger equation
where we have simply used the chain rule on the right hand side.
From the Schrödinger equation itself we deduce that
the adjoint of which gives

53. Some Features of Quantum Mechanical Evolution
Conservation of the Norm - It is also relatively easy to show that quantum mechanical evolution preserves the norm of the state vector, a condition which is obviously important if we wish the total sum of probabilities to be conserved.
Thus, we consider the rate of change of the (squared) length of a vector |𝜓(𝑡)〉 evolving under the Schrödinger equation
where we have simply used the chain rule on the right hand side.
From the Schrödinger equation itself we deduce that
the adjoint of which gives

54. Some Features of Quantum Mechanical Evolution
Conservation of the Norm - It is also relatively easy to show that quantum mechanical evolution preserves the norm of the state vector, a condition which is obviously important if we wish the total sum of probabilities to be conserved.
Thus, we consider the rate of change of the (squared) length of a vector |𝜓(𝑡)〉 evolving under the Schrödinger equation
where we have simply used the chain rule on the right hand side.
From the Schrödinger equation itself we deduce that
the adjoint of which gives

55. Some Features of Quantum Mechanical Evolution
Conservation of the Norm – Making these substitutions in our expression for the time rate of change of the squared norm gives
so that 〈𝜓(𝑡)|𝜓(𝑡)〉=〈𝜓(𝑡₀)|𝜓(𝑡₀)〉 is constant.
This implies that the evolution operator 𝑈=𝑈(𝑡,𝑡₀) is unitary, since 〈𝜓(𝑡)|𝜓(𝑡)〉=〈𝜓(𝑡₀)|𝑈⁺𝑈|𝜓(𝑡₀)〉=〈𝜓(𝑡₀)|𝜓(𝑡₀)〉.
Since this must be true for arbitrary states |𝜓(𝑡₀)〉, it follows that 𝑈⁺𝑈=1.
Thus, the Schrödinger equation leads to what is called a unitary evolution.

56. Some Features of Quantum Mechanical Evolution
Conservation of the Norm – Making these substitutions in our expression for the time rate of change of the squared norm gives
so that 〈𝜓(𝑡)|𝜓(𝑡)〉=〈𝜓(𝑡₀)|𝜓(𝑡₀)〉 is constant.
This implies that the evolution operator 𝑈=𝑈(𝑡,𝑡₀) is unitary, since 〈𝜓(𝑡)|𝜓(𝑡)〉=〈𝜓(𝑡₀)|𝑈⁺𝑈|𝜓(𝑡₀)〉=〈𝜓(𝑡₀)|𝜓(𝑡₀)〉.
Since this must be true for arbitrary states |𝜓(𝑡₀)〉, it follows that 𝑈⁺𝑈=1.
Thus, the Schrödinger equation leads to what is called a unitary evolution.

57. Some Features of Quantum Mechanical Evolution
Conservation of the Norm – Making these substitutions in our expression for the time rate of change of the squared norm gives
so that 〈𝜓(𝑡)|𝜓(𝑡)〉=〈𝜓(𝑡₀)|𝜓(𝑡₀)〉 is constant.
This suggests that the evolution operator 𝑈=𝑈(𝑡,𝑡₀) is unitary, since 〈𝜓(𝑡)|𝜓(𝑡)〉=〈𝜓(𝑡₀)|𝑈⁺𝑈|𝜓(𝑡₀)〉=〈𝜓(𝑡₀)|𝜓(𝑡₀)〉.
Since this must be true for arbitrary states |𝜓(𝑡₀)〉, it follows that 𝑈⁺𝑈=1.
Thus, the Schrödinger equation leads to what is called a unitary evolution.

58. Some Features of Quantum Mechanical Evolution
Conservation of the Norm – Making these substitutions in our expression for the time rate of change of the squared norm gives
so that 〈𝜓(𝑡)|𝜓(𝑡)〉=〈𝜓(𝑡₀)|𝜓(𝑡₀)〉 is constant.
This suggests that the evolution operator 𝑈=𝑈(𝑡,𝑡₀) is unitary, since 〈𝜓(𝑡)|𝜓(𝑡)〉=〈𝜓(𝑡₀)|𝑈⁺𝑈|𝜓(𝑡₀)〉=〈𝜓(𝑡₀)|𝜓(𝑡₀)〉.
Since this must be true for arbitrary states |𝜓(𝑡₀)〉, it follows that 𝑈⁺𝑈=1.
Thus, the Schrödinger equation leads to what is called a unitary evolution.

59. Some Features of Quantum Mechanical Evolution
Conservation of the Norm – Making these substitutions in our expression for the time rate of change of the squared norm gives
so that 〈𝜓(𝑡)|𝜓(𝑡)〉=〈𝜓(𝑡₀)|𝜓(𝑡₀)〉 is constant.
This suggests that the evolution operator 𝑈=𝑈(𝑡,𝑡₀) is unitary, since 〈𝜓(𝑡)|𝜓(𝑡)〉=〈𝜓(𝑡₀)|𝑈⁺𝑈|𝜓(𝑡₀)〉=〈𝜓(𝑡₀)|𝜓(𝑡₀)〉.
Since this must be true for arbitrary states |𝜓(𝑡₀)〉, it follows that 𝑈⁺𝑈=1.
Thus, the Schrödinger equation leads to what is called a unitary evolution.

60. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values - Let us now consider how the mean value associated with an arbitrary observable 𝐴(𝑡) evolves in time.
In general, operators can have an intrinsic time dependence.
As an example consider the potential associated with the application of a spatially uniform sinusoidally-varying electric field, i.e.,
The mean value of this operator
evolves in time, since both the state and the operator itself is changing.
At any instant of time, this mean value gives a measure of the interaction of the system with the external field.

61. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values - Let us now consider how the mean value associated with an arbitrary observable 𝐴(𝑡) evolves in time.
In general, operators can have an intrinsic time dependence.
As an example consider the potential associated with the application of a spatially uniform sinusoidally-varying electric field, i.e.,
The mean value of this operator
evolves in time, since both the state and the operator itself is changing.
At any instant of time, this mean value gives a measure of the interaction of the system with the external field.

62. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values - Let us now consider how the mean value associated with an arbitrary observable 𝐴(𝑡) evolves in time.
In general, operators can have an intrinsic time dependence.
As an example consider the potential associated with the application of a spatially uniform sinusoidally-varying electric field, i.e.,
The mean value of this operator
evolves in time, since both the state and the operator itself is changing.
At any instant of time, this mean value gives a measure of the interaction of the system with the external field.

63. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values - Let us now consider how the mean value associated with an arbitrary observable 𝐴(𝑡) evolves in time.
In general, operators can have an intrinsic time dependence.
As an example consider the potential associated with the application of a spatially uniform sinusoidally-varying electric field, i.e.,
The mean value of this operator
evolves in time, since both the state and the operator itself is changing.
At any instant of time, this mean value gives a measure of the interaction of the system with the external field.

64. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values - Let us now consider how the mean value associated with an arbitrary observable 𝐴(𝑡) evolves in time.
In general, operators can have an intrinsic time dependence.
As an example consider the potential associated with the application of a spatially uniform sinusoidally-varying electric field, i.e.,
The mean value of this operator
evolves in time, since both the state and the operator itself is changing.
At any instant of time, this mean value gives a measure of the interaction of the system with the external field.

65. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values - In general, the mean value of an arbitrary observable 〈𝐴(𝑡)〉=〈𝜓(𝑡)|𝐴(𝑡)|𝜓(𝑡)〉
may have two sources of time dependence:
a part due to the operator itself, and
a part due to the evolution of the state.
We can, however, use the chain rule to write

66. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values - In general, the mean value of an arbitrary observable 〈𝐴(𝑡)〉=〈𝜓(𝑡)|𝐴(𝑡)|𝜓(𝑡)〉
may have two sources of time dependence:
a part due to the operator itself, and
a part due to the evolution of the state.
We can, however, use the chain rule to write

67. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values - In general, the mean value of an arbitrary observable 〈𝐴(𝑡)〉=〈𝜓(𝑡)|𝐴(𝑡)|𝜓(𝑡)〉
may have two sources of time dependence:
a part due to the operator itself, and
a part due to the evolution of the state.
We can, however, use the chain rule to write

68. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values - In general, the mean value of an arbitrary observable 〈𝐴(𝑡)〉=〈𝜓(𝑡)|𝐴(𝑡)|𝜓(𝑡)〉
may have two sources of time dependence:
a part due to the operator itself, and
a part due to the evolution of the state.
We can, however, use the product rule to write

69. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values - In general, the mean value of an arbitrary observable 〈𝐴(𝑡)〉=〈𝜓(𝑡)|𝐴(𝑡)|𝜓(𝑡)〉
may have two sources of time dependence:
a part due to the operator itself, and
a part due to the evolution of the state.
We can, however, use the product rule to write

70. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values –
From our earlier manipulations this can be written
Note: the commutator of 𝐴 and 𝐻 emerges from the 1st and last terms.
Thus, we have derived the following general equation of motion
For the mean value of an arbitrary observable.

71. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values –
From our earlier manipulations this can be written
Note: the commutator of 𝐴 and 𝐻 emerges from the 1st and last terms.
Thus, we have derived the following general equation of motion
For the mean value of an arbitrary observable.

72. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values –
From our earlier manipulations this can be written
Note: the commutator of 𝐴 and 𝐻 emerges from the 1st and last terms.
Thus, we have derived the following general equation of motion
For the mean value of an arbitrary observable.

73. Some Features of Quantum Mechanical Evolution
Evolution of Mean Values –
From our earlier manipulations this can be written
Note: the commutator of 𝐴 and 𝐻 emerges from the 1st and last terms.
Thus, we have derived the following general equation of motion
For the mean value of an arbitrary observable.

74. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - As an interesting application of the use of the equations of motion for the mean value of an observable, consider the motion of a particle under the influence of a force
derivable from a scalar potential.
Quantum mechanically, this corresponds to the usual Hamiltonian
Let us now consider how the mean values and associated with the corresponding quantum mechanical observables change in time.

75. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - As an interesting application of the use of the equations of motion for the mean value of an observable, consider the motion of a particle under the influence of a force
derivable from a scalar potential.
Quantum mechanically, this corresponds to the usual Hamiltonian
Let us now consider how the mean values and associated with the corresponding quantum mechanical observables change in time.

76. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - As an interesting application of the use of the equations of motion for the mean value of an observable, consider the motion of a particle under the influence of a force
derivable from a scalar potential.
Quantum mechanically, this corresponds to the usual Hamiltonian
Let us now consider how the mean values and associated with the corresponding quantum mechanical observables change in time.

77. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - First, we examine the equation of motion for the position operator , which being independent of time leads to the equation of motion
This leads us to evaluate
Since V is a function of R, the second commutator vanishes.
The 𝑥 component of the first commutator is

78. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - First, we examine the equation of motion for the position operator , which being independent of time leads to the equation of motion
This leads us to evaluate
Since V is a function of R, the second commutator vanishes.
The 𝑥 component of the first commutator is

79. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - First, we examine the equation of motion for the position operator , which being independent of time leads to the equation of motion
This leads us to evaluate
Since V is a function of R, the second commutator vanishes.
The 𝑥 component of the first commutator is

80. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - First, we examine the equation of motion for the position operator , which being independent of time leads to the equation of motion
This leads us to evaluate
Since V is a function of R, the second commutator vanishes.
The 𝑥 component of the first commutator is

81. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - First, we examine the equation of motion for the position operator , which being independent of time leads to the equation of motion
This leads us to evaluate
Since V is a function of R, the second commutator vanishes.
The 𝑥 component of the first commutator is

82. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - First, we examine the equation of motion for the position operator , which being independent of time leads to the equation of motion
This leads us to evaluate
Since V is a function of R, the second commutator vanishes.
The 𝑥 component of the first commutator is

83. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - First, we examine the equation of motion for the position operator , which being independent of time leads to the equation of motion
This leads us to evaluate
Since V is a function of R, the second commutator vanishes.
The 𝑥 component of the first commutator is

84. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - As a vector operator relation, therefore, we have the result that
[R,P²]=2iℏP,
which we can put back into the equation of motion for 〈R(t)〉 to obtain
Thus, we find that
which, brackets aside, looks like its classical counterpart.

85. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - As a vector operator relation, therefore, we have the result that
[R,P²]=2iℏP,
which we can put back into the equation of motion for 〈R(t)〉 to obtain
Thus, we find that
which, brackets aside, looks like its classical counterpart.

86. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - As a vector operator relation, therefore, we have the result that
[R,P²]=2iℏP,
which we can put back into the equation of motion for 〈R(t)〉 to obtain
Thus, we find that
which, brackets aside, looks like its classical counterpart.

87. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - As a vector operator relation, therefore, we have the result that
[R,P²]=2iℏP,
which we can put back into the equation of motion for 〈R(t)〉 to obtain
Thus, we find that
which, brackets aside, looks like its classical counterpart.

88. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem - As a vector operator relation, therefore, we have the result that
[R,P²]=2iℏP,
which we can put back into the equation of motion for 〈R(t)〉 to obtain
Thus, we find that
which, brackets aside, looks like its classical counterpart.

89. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Thus, as in classical mechanics the mean velocity equals the mean momentum divided by the mass.
In a similar fashion we can compute the equation of motion for the mean momentum,
which leads us to evaluate
Now the kinetic energy term disappears, but the potential energy term does not, since it is a function of the operator 𝑅 , which does not commute with 𝑃 .

90. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Thus, as in classical mechanics the mean velocity equals the mean momentum divided by the mass.
In a similar fashion we can compute the equation of motion for the mean momentum,
which leads us to evaluate
Now the kinetic energy term disappears, but the potential energy term does not, since it is a function of the operator 𝑅 , which does not commute with 𝑃 .

91. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Thus, as in classical mechanics the mean velocity equals the mean momentum divided by the mass.
In a similar fashion we can compute the equation of motion for the mean momentum,
which leads us to evaluate
Now the kinetic energy term disappears, but the potential energy term does not, since it is a function of the operator 𝑅 , which does not commute with 𝑃 .

92. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Thus, as in classical mechanics the mean velocity equals the mean momentum divided by the mass.
In a similar fashion we can compute the equation of motion for the mean momentum,
which leads us to evaluate
Now the kinetic energy term disappears, but the potential energy term does not, since it is a function of the operator 𝑅 , which does not commute with 𝑃 .

93. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Since we have not specified the exact functional form of the potential energy function, it is convenient to work in a representation in which V is diagonal, namely the position representation.
In the position representation we can write
where

94. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Since we have not specified the exact functional form of the potential energy function, it is convenient to work in a representation in which V is diagonal, namely the position representation.
In the position representation we can write
where

95. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Since we have not specified the exact functional form of the potential energy function, it is convenient to work in a representation in which V is diagonal, namely the position representation.
In the position representation we can write
where

96. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Since we have not specified the exact functional form of the potential energy function, it is convenient to work in a representation in which V is diagonal, namely the position representation.
In the position representation we can write
where

97. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Since we have not specified the exact functional form of the potential energy function, it is convenient to work in a representation in which V is diagonal, namely the position representation.
In the position representation we can write
where

98. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Since we have not specified the exact functional form of the potential energy function, it is convenient to work in a representation in which V is diagonal, namely the position representation.
In the position representation we can write
where

99. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Thus, we make the identification
where is clearly the force operator.
Using this result in the equation of motion we find that
Thus, the equations of motion for the position and momentum operators can be written
which looks like Newton's equations, aside from the taking of expectation values.

100. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Thus, we make the identification
where is clearly the force operator.
Using this result in the equation of motion we find that
Thus, the equations of motion for the position and momentum operators can be written
which looks like Newton's equations, aside from the taking of expectation values.

101. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Thus, we make the identification
where is clearly the force operator.
Using this result in the equation of motion we find that
Thus, the equations of motion for the position and momentum operators can be written
which looks like Newton's equations, aside from the taking of expectation values.

102. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
Thus, we make the identification
where is clearly the force operator.
Using this result in the equation of motion we find that
Thus, the equations of motion for the position and momentum operators can be written
which looks like Newton's equations, aside from the taking of expectation values.

103. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
These classically familiar-looking expressions are referred to as Ehrenfest's equations of motion for the mean values.
Their interpretation requires a little care.
It might be expected, for example, that these equations imply that if the initial mean values were equal to those of some hypothetical classical system with the same potential, so that 〈 𝑅 (0)〉= 𝑟 (0) and 〈 𝑃 (0)〉= 𝑝 (0), then as both systems evolved the mean values 〈 𝑅 (𝑡)〉 and 〈 𝑃 (𝑡)〉 for the quantum particle would simply follow the corresponding classical trajectory 𝑟 (𝑡) and 𝑝 (𝑡).
This is, however, not generally the case.

104. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
These classically familiar-looking expressions are referred to as Ehrenfest's equations of motion for the mean values.
Their interpretation requires a little care.
It might be expected, for example, that these equations imply that if the initial mean values were equal to those of some hypothetical classical system with the same potential, so that 〈 𝑅 (0)〉= 𝑟 (0) and 〈 𝑃 (0)〉= 𝑝 (0), then as both systems evolved the mean values 〈 𝑅 (𝑡)〉 and 〈 𝑃 (𝑡)〉 for the quantum particle would simply follow the corresponding classical trajectory 𝑟 (𝑡) and 𝑝 (𝑡).
This is, however, not generally the case.

105. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
These classically familiar-looking expressions are referred to as Ehrenfest's equations of motion for the mean values.
Their interpretation requires a little care.
It might be expected, for example, that these equations imply that if the initial mean values were equal to those of some hypothetical classical system with the same potential, so that 〈 𝑅 (0)〉= 𝑟 (0) and 〈 𝑃 (0)〉= 𝑝 (0), then as both systems evolved the mean values 〈 𝑅 (𝑡)〉 and 〈 𝑃 (𝑡)〉 for the quantum particle would simply follow the corresponding classical trajectory 𝑟 (𝑡) and 𝑝 (𝑡).
This is, however, not generally the case.

106. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
These classically familiar-looking expressions are referred to as Ehrenfest's equations of motion for the mean values.
Their interpretation requires a little care.
It might be expected, for example, that these equations imply that if the initial mean values were equal to those of some hypothetical classical system with the same potential, so that 〈 𝑅 (0)〉= 𝑟 (0) and 〈 𝑃 (0)〉= 𝑝 (0), then as both systems evolved the mean values 〈 𝑅 (𝑡)〉 and 〈 𝑃 (𝑡)〉 for the quantum particle would simply follow the corresponding classical trajectory 𝑟 (𝑡) and 𝑝 (𝑡).
This is, however, not generally the case.

107. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
A careful comparison of the two cases shows, that for the quantum mean values to follow the corresponding classical trajectories, the quantum equation of motion would have to be
rather than
These are usually not the same. It is easy to show, however that these are the same, i.e., that, at each instant or a free particle (for which the force vanishes), a constant force, or for a linear restoring force, i.e., a harmonic oscillator.

108. Some Features of Quantum Mechanical Evolution
Ehrenfest’s Theorem
A careful comparison of the two cases shows, that for the quantum mean values to follow the corresponding classical trajectories, the quantum equation of motion would have to be
rather than
These are usually not the same. It is easy to show, however that these are the same, i.e., that, at each instant or a free particle (for which the force vanishes), a constant force, or for a linear restoring force, i.e., a harmonic oscillator.

109. In this lecture, we finished our discussion of the 3rd postulate, and stated and began discussing the 4th postulate of our general formulation of quantum mechanics, which gives Schrödinger's equation of motion governing the evolution of the state vector.
We saw to express Schrödinger's equation in different representations, and discussed some general features of quantum mechanical evolution, including its linear, deterministic nature (introducing the unitary evolution operator in the process), the fact that it conserves the norm of the state vector, and we derived the equations of motion for the mean value of arbitrary observables.
As an example of the latter, we derived classical-resembling equations of motion for the mean value of position and momentum, usually referred to as Eherenfest’s theorem.
In the next lecture we explore the consequences of Schrödinger's equation for systems with a time-independent Hamiltonian.

110. In this lecture, we finished our discussion of the 3rd postulate, and stated and began discussing the 4th postulate of our general formulation of quantum mechanics, which gives Schrödinger's equation of motion governing the evolution of the state vector.
We saw to express Schrödinger's equation in different representations, and discussed some general features of quantum mechanical evolution, including its linear, deterministic nature (introducing the unitary evolution operator in the process), the fact that it conserves the norm of the state vector, and we derived the equations of motion for the mean value of arbitrary observables.
As an example of the latter, we derived classical-resembling equations of motion for the mean value of position and momentum, usually referred to as Eherenfest’s theorem.
In the next lecture we explore the consequences of Schrödinger's equation for systems with a time-independent Hamiltonian.

111. In this lecture, we finished our discussion of the 3rd postulate, and stated and began discussing the 4th postulate of our general formulation of quantum mechanics, which gives Schrödinger's equation of motion governing the evolution of the state vector.
We saw to express Schrödinger's equation in different representations, and discussed some general features of quantum mechanical evolution, including its linear, deterministic nature (introducing the unitary evolution operator in the process), the fact that it conserves the norm of the state vector, and we derived the equations of motion for the mean value of arbitrary observables.
As an example of the latter, we derived classical-resembling equations of motion for the mean value of position and momentum, usually referred to as Ehrenfest’s theorem.
In the next lecture we explore the consequences of Schrödinger's equation for systems with a time-independent Hamiltonian.

112. In this lecture, we finished our discussion of the 3rd postulate, and stated and began discussing the 4th postulate of our general formulation of quantum mechanics, which gives Schrödinger's equation of motion governing the evolution of the state vector.
We saw to express Schrödinger's equation in different representations, and discussed some general features of quantum mechanical evolution, including its linear, deterministic nature (introducing the unitary evolution operator in the process), the fact that it conserves the norm of the state vector, and we derived the equations of motion for the mean value of arbitrary observables.
As an example of the latter, we derived classical-resembling equations of motion for the mean value of position and momentum, usually referred to as Ehrenfest’s theorem.
In the next lecture we explore the consequences of Schrödinger's equation for systems with a time-independent Hamiltonian.