In this report, we start by defining key aspects of classical Lagrangian mechanics including the principle of least action and how one can use this to derive the Euler-Lagrange equations. Momentum and Conservation laws shall also be introduced, deriving relations between position, momenta and the Lagrangian of a given system. Following this, we develop our study of classical mechanics further using Legendre transforms on the Euler-Lagrange equation and our conservation laws to define Hamiltonian mechanics. In our new notation, we use Poisson brackets when evaluating the rate of change of a classical observable. Next, we cross to quantum mechanics, giving some definitions which shall be used for later discussion. We then state and prove the Ehrenfest theorem, from which we draw our first correspondence between classical and quantum mechanics, most notably between the Poisson bracket and the commutator. Furthermore, the Ehrenfest theorem applied to operators of position and momentum shows a further correspondence with classical results. Finally, we take an example of the simple harmonic oscillator, using both classical and quantum methods to solve for this system and comment on the similarities and differences between the results graphically and qualitatively.
1. Introduction
In life, one would take the shortest route to get to where they are going and therefore expel the least possible amount of energy and the particles in our universe act no different. This principle is described by Lagrangian mechanics, developed by Joseph-Louis Lagrange in the late 18th century. [1] An extension from Newtonian mechanics, Lagrangian methods were created in order to be able to work with more complicated systems, using a new coordinate system which can take account for any system constraints. The key principle involves minimizing the value of an integral in order to derive the equations of motion of a particle. These equations can be used to unlock many new ideas and create another equivalent branch of classical mechanics.
The formalism of Hamiltonian mechanics was made in 1833 by William Hamilton. It predicts the same outcomes as Lagrangian methods, except providing a deeper abstract understanding of the theory behind it. Similar to Lagrangian mechanics it is focused around a single function, called the Hamiltonian, this function is sum of the kinetic and potential energies of the system. [2] The Hamiltonian can be related to the time derivatives of position and momentum by Hamilton’s equations of motion. This formulation of mechanics was critically important in the discovery of statistical branches of mechanics, such as quantum mechanics.
Quantum mechanics describes our universe at the smallest scale of energy levels where certain quantities can only take discrete values. It models every particle as having a wave function that exists in a given state. It was developed in the early 20th century by a series of different mathematicians such as, Richard Feynman and Paul Dirac. The particles are said to have wavelike properties and behave in accordance with the Schrödinger wave equation. This equation contains the quantum Hamiltonian which again is the sum of the kinetic and potential energies as in classical mechanics. It makes predictions by assigning probabilities to the expectation values of a physical observable represented by quantum operators, and can be used to evaluate classical systems such as the harmonic oscillator.
There is a correspondence between the three methods of mechanics. We shall study the similarities and differences in the results that these approaches produce and how they can be related mathematically.
2. Lagrangian Mechanics
We begin by exploring a re-formulation of Newtonian mechanics developed by Joseph-Louis Lagrange called Lagrangian Mechanics. For a given physical system we require equations of motion which contain variables as functions of time, in order to pinpoint the location of an object or particle at any given time. The majority of physical systems are not free, and motion is restricted by properties of the system. These systems are called constrained systems. [2]
Definition 2.1- A constrained system is a system that is subject to either [3]:
Geometric constraints: factors which impose some limit to the position of an object. [2]
Kinematical constraints: factors which describe how the velocity of a particle behaves. [2]
Definition 2.2- A function for which the integral can be computed is said to be integrable. [4]
Definition 2.3- A system is said to be holonomic if it has only geometrical or integrable kinematical constraints. [2]
Since the classical Newtonian equations using Cartesian coordinates do not have these constraints we must find a new coordinate system to work with.
Definitions 2.3- Let S be a system and x=〖(x〗_1,…,x_n) be a set of independent variables. If the position of every particle in S can be written as a function of these variables we say that x=〖(x〗_1,…,x_n) are a set of generalised coordinates for S. The time derivatives 〖x ̇=(x ̇〗_1,…,x ̇_n) of these generalised coordinates are called the generalised velocities of S. [2][3]
Definition 2.4- Let S be a holonomic system. The number of degrees of freedom of S is the number of generalised coordinates x=〖(x〗_1,…,x_n) required to describe the configuration of S. The number of degrees of freedom of a system is equal to the number of equations of motion needed to find the motion of the system. [2]
Definition 2.5- Let S be a holonomic system with generalised coordinates. Then the Lagrangian function L is,
█(L (x,x ̇,t)= T(x,x ̇,t)-V(x,t).#(2.1) )
Here, our Lagrangian function is dependent on the set of generalized coordinates x, the generalised velocities x ̇, and time t. [2]
3. Calculus of Variations
The method of calculus of variations is used to find the stationary values on a path, curve, surface, etc. of a given function with fixed end points by using an integral.
Definition 3.1- Let F(σ) be a real valued function, which we call an action of function f(σ) for [a,b]. [2] We can write this in the form of an integral,
█(F(σ)≔∫_a^b▒〖f(σ,σ ̇,t) dt〗.#(3.1) )
Definition 3.2- The correct path of motion of a mechanical system with holonomic constraints and conservative external forces, from time t_1 to t_2, is the stationary solution of the action. This path satisfies Lagrange’s equations of motion and is called the principle of least action. [5]
Lemma 3.3- (Euler-Lagrange Lemma) [6] If α(σ) is a continuous function on [a,b], and
█(∫_a^b▒〖α(σ)ρ(σ) dσ=0〗,#(3.2) )
for all continuously differentiable functions ρ(σ) which satisfy ρ(σ_a )=ρ(σ_b )=0, then,
█(α(σ)≡0 on [a,b].#(3.3) )
Proof. A proof of the Euler-Lagrange Lemma can be found in [6] pg.189.
Theorem 3.4- Suppose the function x(t) minimises the action I(x), then it must satisfy the following equation on [t_1,t_2],
█(d/dt (∂f/(∂x ̇ ))-∂f/∂x=0.#(3.4) )
This is called the Euler-Lagrange equation. [2]
Proof. Following similar derivations as in [6] and [7], we start with an action I(x), where f is a given function of x(t),x ̇(t) and t. Let x(t) be a twice differentiable function, with fixed at en
d points, x(t_1 )=x_1,x(t_2 )=x_2. Leaving the following,
█(I(x)=∫_(t_1)^(t_2)▒〖f(x(t),x ̇(t
),t) dt〗.#(3.5) )
We want to find the extremum points of the action in order to find the value of x(t) such that I(x) is the required minimum.
We begin by assuming that x(t) is the function that minimises our action and that satisfies the required boundary conditions on x. Now, we introduce a continuous twice differentiable function ρ(t) defined on [t_1,t_2], which satisfies 〖ρ(t〗_1)=ρ(t_2 )=0. Define,
█(x ̃(t)≔x(t)+ερ(t),#(3.6) )
where ε is an arbitrarily small real parameter. We set,
█(φ(ε)≔I(x ̃ )=I[x(t)+ερ(t)].#(3.7) )
We want to find the extremum of I(x ̃ ) at x, this means that x is a stationary function for I(x ̃), and for all ρ(t), we require
█(dφ(ε=0)/dε=0.#(3.8) )
Differentiating φ(ε) with respect to parameter ε,
█(dφ/dε=d/dε ∫_(t_1)^(t_2)▒〖f(x+ερ,x ̇+ερ ̇,t) dt〗.#(3.9) )
By a property of Calculus, we bring the d/dε into the integral giving,
█(dφ/dε=∫_(t_1)^(t_2)▒∂/∂ε f(x+ερ,x ̇+ερ ̇,t) dt,#(3.10) )
and using the chain rule to evaluate the integrand,
█(dφ/dε=∫_(t_1)^(t_2)▒(∂f/(∂x ̃ ) (dx ̃)/dε+∂f/(∂x ̃ ̇ ) (dx ̃ ̇)/dε)dt.#(3.11) )
Applying our definition of x ̃(t), it is clear to see that (dx ̃)/dε=ρ and similarly that (dx ̃ ̇)/dε=ρ ̇, hence,
█( dφ/dε =∫_(t_1)^(t_2)▒( ∂f/(∂x ̃ ) ρ+∂f/(∂x ̃ ̇ ) ρ ̇ )dt.#(3.12) )
Integrating the term containing the ρ ̇ using the integration by parts formula, we name u=∂f/(∂x ̃ ̇ ),v=ρ and u ̇=d/dt (∂f/(∂x ̃ ̇ )),v ̇=ρ ̇.
█(∫_(t_1)^(t_2)▒(∂f/(∂x ̃ ̇ ) ρ ̇ )dt=[∂f/(∂x ̃ ̇ ) ρ] t_2¦t_1 -∫_(t_1)^(t_2)▒〖d/dt (∂f/(∂x ̃ ̇ ))ρ dt,〗#(3.13) )
and our equation (3.12) becomes,
█(dφ/dε=[∂f/(∂x ̃ ̇ ) ρ] t_2¦t_1 +∫_(t_1)^(t_2)▒〖[ ∂f/(∂x ̃ ) ρ-d/dt (∂f/(∂x ̃ ̇ ))ρ]dt.〗#(3.14) )
Evaluate the first term of (3.14) using ρ(t_1 )=ρ(t_2 )=0,
█([∂f/(∂x ̃ ̇ ) ρ] t_2¦t_1 =∂f/(∂x ̃ ̇ ) [ρ(t_2 )-ρ(t_1 )]=∂f/(∂x ̃ ̇ ) [0-0]=0.#(3.15) )
Substituting into equation (3.14) leaves,
█(dφ/dε= 0-∫_(t_1)^(t_2)▒〖[ d/dt (∂f/(∂x ̃ ̇ ))-∂f/(∂x ̃ )]ρ dt〗=0.#(3.16) )
By taking ε=0, we arrive at x ̃(t)=x(t) and by factoring out a (-1) we are left with the integral,
█(dφ(0)/dε=∫_(t_1)^(t_2)▒〖[ d/dt (∂f/(∂x ̇ ))-∂f/∂x]ρ dt〗=0.#(3.17) )
Finally, applying Lemma 3.3 we see our required result,
█(d/dt (∂f/(∂x ̇ ))-∂f/∂x=0.#(3.18) )
This is the Euler-Lagrange equation for x(t). It can be used to solve our problems involving the least action principle. The reversal of the argument also shows that if x(t) satisfies (3.18) then x(t) is an extremum of I(x). Hence,
█(x(t) is and extremum of I(x)⟺ d/dt (∂f/(∂x ̇ ))-∂f/∂x=0.#(3.19) )
Definition 3.5- If S is a holonomic system with generalised coordinates x and Lagrangian L= L (x,x ̇,t). Then the equations of motion of the system can be written in the following form,
█(d/dt (∂L/(∂x ̇ ))-∂L/∂x=0.#(3.20) )
We call this Lagrange’s equations of motion. [2]
The Lagrangian approach to mechanics is to find the extrema minimum value of an integral in order to derive the equations of motion for that system.
4. Momentum and Conservation Laws
Let S be a holonomic system with a set of generalised coordinates and the Euler-Lagrange equations of motion with n degrees of freedom. The Lagrangian for this system is clearly be given by,
█(L=L(x,x ̇,t)=L(x_1,…,x_n,x ̇_1,…,x ̇_n,t).#(4.1) )
Definition 4.1- If a generalised coordinate x_i of a mechanical system S is not contained in the Lagrangian L such that,
█(L=L(x_1,…,x_(i-1),x_(i+1),…,x_n,x ̇_1,…,x ̇_n,t).#(4.2) )
Then we call x_i an ignorable coordinate. [8][9]
At an ignorable coordinate x_i the Euler-Lagrange equation states,
█(d/dt (∂L/(∂x ̇_i ))-∂L/(∂x ̇_i )=0,i=1,2,….,n.#(4.3) )
Here, the term ∂L/(∂x ̇_i )=0, because L has no x_i dependence, hence,
█(d/dt (∂L/(∂x ̇_i ))=0 ⇔ dL/(dx ̇_i )=const.#(4.4) )
Definition 4.2- Consider a holonomic system S with Lagrangian L=L (x,x ̇,t), such that we can define a p_x,
█(p_x≔∂L/(∂x ̇ )=mx ̇,#(4.6) )
which we call the momentum of a free particle. Now say S is a system described by generalised coordinates x=x_1,…,x_n. One can define quantities p_k as,
█(p_k≔∂L/(∂x ̇_k ),(1≤k≤n).#(4.7) )
This is called the generalised momenta for coordinate x_k. [10]
This concept of generalised momenta is useful, because it can be substituted into equation (4.3) giving, a further simplified Euler-Lagrangian equation such that p_k=constant. Therefore, this shows that the generalised momentum for the ignorable coordinate, x_i, is constant.
We can also find the time derivative of this generalised momenta simply using (4.7) in the Euler-Lagrange equation (3.20).
█(d/dt (∂L/(∂x ̇_k ))-∂L/(∂x_k )=d/dt (p_k )-∂L/(∂x_k )=0.#(4.8) )
Then using common notation p ̇_k=d/dt (p_k ) one can see the result,
█(p ̇_k=∂L/(∂x_k ),(1≤k≤n).#(4.9) )
Theorem 4.3- For all ignorable coordinates, x_i∈ {x_1,…,x_n}, the generalised momenta are not time dependent; this is called conserved momentum. [11]
The conservation laws in Lagrangian mechanics are more general than in Newtonian mechanics. Therefore, the Lagrangian can also be used to prove the conservation laws that were proved previously in Newtonian mechanics.
5. Hamiltonian Mechanics
We shall now introduce Hamiltonian mechanics and see how it can be derived from the Lagrangian mechanics that we have already seen. The Hamiltonian formulation adds no new physics to what we have already learnt, however it does provide us with a pathway to the Hamilton-Jacobi equations and quantum mechanics.
Definition 5.1- An active variable is the one that is transformed by a transformation between two functions. The two functions may also have dependence on other variables that are not part of the transformation, these are called passive variables. [2]
Definition 5.2- We have the variables a= (a_1,…,a_n) which are functions of the active variables b=(b_1,…,b_n ) and passive variables c= (c_1,…,c_m ). Then a can be defined by the following formula,
█(a=∇_b F(b,c),#(5.1) )
where F is a given function of b and c. With inverse,
█(b=∇_a G(a,c).#(5.2) )
The function G
is related to F by the formula,
█(G(a,c)=b∙a-F(b,c),#(5.3) )
where b∙a is the standard vector dot product (b∙a=b_1 a_1+⋯+b_n a_n). Moreover, the derivatives of F and G with res
pect to the passive variables c are given by,
█(∇_c F(b,c)=-∇_c G(a,c).#(5.4) )
The relationship between the two functions F and G is symmetric and is said to be the Legendre Transform of the other. [2]
Let S be a Lagrangian system with n degrees of freedom and generalised coordinates x=〖(x〗_1,…,x_n). Then the Euler-Lagrange equations of motion for S are,
█(d/dt (∂L/(∂x ̇ ))-∂L/∂x=0,(1≤k≤n),#(5.5) )
where L=L (x,x ̇,t) is the Lagrangian of the system. We now want to convert this set of n second order ODE’s into Hamiltonian form in terms of unknowns x=〖(x〗_1,…,x_n) and p=(p_1,…,p_n), where {p_k} are the generalised momenta of S (4.7). These can be written in vector form,
█(p=∇_x ̇ L(x,x ̇,t).#(5.6) )
We want to eliminate the velocities x ̇ from the Lagrangian. To do this we use the Legendre transforms. Leaving us with,
█(x ̇=∇_p H(x,p,t).#(5.7) )
This leads us to the definition of the Hamiltonian function.
Definition 5.3- The function H(x,p,t), which is the Legendre transform of the Lagrangian function L (x,x ̇,t) must obey the following equation,
█(H(x,p,t)=(x ) ̇p-L(x,x ̇,t).#(5.8) )
where H(x,p,t) is called the Hamiltonian function of S. [2]
Definition 5.4- We can now use (5.4) to form a relation between H and L with respect to the passive variable x.
█(∇_x L(x,x ̇,t)=-∇_x H(x,p,t).#(5.9) )
Using this relation, we can transform the Lagrange equations into Hamilton’s equations. Take (4.9) which has equivalent vector form,
█(p ̇=∇_x L(x,x ̇,t).#(5.10) )
Which can be transformed into Hamiltonian notation by using (5.9) giving,
█(p ̇=-∇_x H(x,p,t).#(5.11) )
This leaves us with the two transformed Lagrange equations (5.7) and (5.11), known as Hamilton’s equations. [2] Which have expanded form,
█(x ̇_k=∂H/(∂p_k ),p ̇_k=-∂H/(∂x_k ),(1≤k≤n).#(5.12) )
Definition 5.5- Let u (x_k,p_k ) and v (x_k,p_k) be two Classical observables. We define Poisson bracket {u,v} as, [12]
█({u,v}=∑_(k=1)^n▒(∂u/(∂x_k ) ∂v/(∂p_k )-∂u/(∂p_k ) ∂v/(∂x_k )) .#(5.13) )
Let S be a system with n degrees of freedom and generalised coordinates x_k and momenta p_k. In the system, we have an observable A (x_k,p_k,t). Calculating its time derivative, we have
█(dA/dt=∑_(k=1)^n▒(∂A/(∂x_k ) x ̇_k+∂A/(∂p_k ) p ̇_k ) +∂A/∂t.#(5.14) )
Using the Hamilton’s equations in (5.12) we can replace x ̇_k and p ̇_k leaving,
█(dA/dt=∑_(k=1)^n▒(∂A/(∂x_k ) ∂H/(∂p_k )-∂A/(∂x_k ) ∂H/(∂q_k )) +∂A/∂t.#(5.15) )
Now applying the definition of the Poisson bracket, we can concisely write the first term,
█(dA/dt={A,H}+∂A/∂t.#(5.16) )
We shall refer to this result when looking at the Ehrenfest theorem. [12]
Fundamentally, there is no difference between Lagrangian and Hamiltonian mechanics, both methods will give equivalent solutions for the evolution with respect to time of a physical system. The Lagrangian uses the principle of least action to solve a problem and gives us insight into symmetries of the system. Whereas, for the Hamiltonian it simpler to calculate the time evolutions given a set of generalised coordinates and momenta for the system.
6. Classical Limit and Correspondence Principle
Quantum mechanics is built upon an analogy with the Hamiltonian classical mechanics. Using the theory of Hamiltonian mechanics to give statistical interpretations of a physical system on a macroscopic scale. [12]
The theory of quantum mechanics is built upon a set of postulates. [13] In brief summary, they state that:
The state of a particle can be represented by a vector |├ ψ (x,t)⟩ in the Hilbert space.
The independent variables x and p from classical interpretations become hermitian operators x ̂ and p ̂. In general, observables from classical mechanics become operators in quantum mechanics.
If we study a particle in state |├ ψ (x,t)⟩, a measurement of observable A ̂ will give an eigenvalue a_i and a probability of yielding this state ∝ |⟨a_i │ψ⟩|^2.
The state vector |├ ψ (x,t)⟩ obeys the Schrödinger equation:
█(iℏ d/dt ├ ├|ψ┤ (x,t)⟩= (p ̂^2/2m+V(x))├ ├|ψ(x,t)┤⟩= H ̂├ ├|ψ(x,t)┤⟩,#(6.1) )
where H ̂ is the Quantum Hamiltonian Operator, equal to the sum of kinetic and potential energies. [9]
Definition 6.1- The expectation value of a given observable, represented by operator A ̂ is the average value of the observable over the ensemble. [13] Say every particle is in the state ├ |ψ⟩ then,
█(〈A ̂ 〉=〈ψ|A ̂ |ψ〉.#(6.2) )
Definition 6.2- Let A ̂ be a quantum operator representing a physical observable. We say A ̂ is a Hermitian operator if,
█(A ̂=A ̂^†.#(6.3) )
Where A ̂^† is the adjoint of the operator (definition can be found in [13] pg. 22). An example of a Hermitian operator is the Hamiltonian operator. [13]
Definition 6.3- The commutator of two quantum operators is defined as,
█([A ̂,B ̂ ]≡A ̂B ̂-B ̂A ̂.#(6.4) )
If [A ̂,B ̂ ]=0 then we say the operators commute. It is also noted that the order of the operators can change the result, [A ̂,B ̂ ]=-[B ̂,A ̂ ] and that in general, A ̂B ̂≠B ̂A ̂. [14]
Theorem 6.4- Let A ̂,B ̂ and C ̂, be quantum operators then,
█([A ̂,B ̂C ̂ ]=[A ̂,B ̂ ] C ̂+B ̂[A ̂,C ̂ ].#(6.5) )
Proof. First, we apply the definition of the commutator (6.4). We then add zero (+B ̂A ̂C ̂-B ̂A ̂C ̂) and simplify using the Definition 6.3,
█([A ̂,B ̂C ̂ ]=A ̂(B ̂C ̂ )-(B ̂C ̂ ) A ̂#(6.6) )
█( =A ̂B ̂C ̂-B ̂C ̂A ̂+B ̂A ̂C ̂-B ̂A ̂C ̂#(6.7) )
█( =(A ̂B ̂-B ̂A ̂ ) C ̂+B ̂(A ̂C ̂-C ̂A ̂ )#(6.8) )
█( =[A ̂,B ̂ ] C ̂+B ̂[A ̂,C ̂ ].#(6.9) )
Two commutation relations which we shall use in later discussion are,
█([x ̂,p ̂ ]=iℏ, [p ̂,V(x)]=-iℏ dV(x)/dx.#(6.10) )
The proofs for these can be found in [13] and [15].
Theorem 6.5- The generalized Ehrenfest theorem for expectation value of a quantum operator A ̂=A ̂ (x,t) is
█(d/dt 〈A ̂ 〉=1/iℏ 〈[A ̂,H ̂ ]〉+〈(∂A ̂)/∂t〉,#(6.11) )
where H ̂ is the Hamiltonian operator. [15]
Proof. We start by applying the definition of the expectation value of a general operator (6.14),
█( d/dt 〈A ̂ 〉=d/dt 〈ψ(x,t)|A ̂ |ψ(x,t)〉.#(6.12) )
Taking the derivative into the expectation value gives,
█(= 〈∂ψ(x,t)/∂t |A ̂ |ψ(x,t)〉+〈ψ(x,t)|(∂A ̂)/∂t|ψ(x,t)〉+〈ψ(x,t)|A ̂ | ∂ψ(x,t)/∂t〉.#(6.
13) )
We can now simply evaluate the time derivatives of ψ in the bras and kets by rearranging the Schrödinger equation (6.1).
█(├ ├|∂ψ/∂t┤⟩=d/dt ├ ├|ψ┤⟩=1/
iℏ H ̂├ ├|ψ┤⟩,#(6.14) )
and similarly using the fact H ̂ is Hermitian.
█(⟨├ ∂ψ/∂x┤|┤=(├ ├|∂ψ/∂t┤⟩)^†=(1/iℏ H ̂├ ├|ψ┤⟩)^†=-1/iℏ ⟨├ ψ┤|┤ H ̂^†=-1/iℏ ⟨├ ψ┤|┤ H ̂ .#(6.15) )
Using results (6.14) and (6.15) in (6.13) we have,
█(d/dt 〈A ̂ 〉=-1/iℏ 〈ψ(x,t)|H ̂A ̂ |ψ(x,t)〉+〈ψ(x,t)|(∂A ̂)/∂t|ψ(x,t)〉+1/iℏ 〈ψ(x,t)|A ̂H ̂ |ψ(x,t)〉.#(6.16) )
We can now combine the first and third term in (6.16) using the commutation relation (6.4).
█(d/dt 〈A ̂ 〉=1/iℏ 〈ψ(x,t)|[A ̂,H ̂ ]|ψ(x,t)〉+〈ψ(x,t)|(∂A ̂)/∂t|ψ(x,t)〉.#(6.17) )
Finally, we apply the definition of expectation value (6.2) on both terms in (6.17) and we are left with the Ehrenfest theorem for a general quantum operator (6.11).
The Ehrenfest Theorem corresponds structurally to a result in classical mechanics. If we take a classical observable A which depends on set of generalised coordinates x_k and momenta p_k, then calculate its rate of change we see as shown for (5.16) that,
█(dA/dt={A,H}+∂A/∂t.#(6.18) )
From this we can see an immediate correspondence between the classical Poisson bracket (5.13) and the quantum commutator (6.4),
█({A,H}→1/iℏ [A ̂,H ̂ ].#(6.19) )
Here we can see that the classical Poisson bracket is equivalent to the quantum commutator with a factor of iℏ present. This is put in practice when computing the commutator [x ̂,p ̂ ]=iℏ, in this case the Poisson bracket {x,p}=1, hence satisfying our derived relation.
Now, we look at some key results from the Ehrenfest theorem and how they can help us find further correspondence between classical and quantum mechanics.
Example 6.6- In this example we shall look at a specific case of the Ehrenfest theorem where we set A ̂=x ̂ the position operator. [15] For a Hamiltonian,
█(H ̂=p ̂^2/2m+V(x).#(6.20) )
We begin by substituting x ̂ into (6.11),
█(d/dt 〈x ̂ 〉=1/iℏ 〈[x ̂,H ̂ ]〉+〈(∂x ̂)/∂t〉.#(6.21) )
It is clear to see that the second term in this equation disappears as x ̂ has no time dependence. We now use our Hamiltonian to expand the commutator.
█(d/dt 〈x ̂ 〉=1/iℏ 〈[x ̂,p ̂^2/2m+V(x)]〉.#(6.22) )
Here x ̂ commutes with V(x) (Definition 6.3) so we are only left with the commutator [x ̂,p ̂^2 ].
█(d/dt 〈x ̂ 〉=1/2imℏ 〈[x ̂,p ̂^2 ]〉.#(6.23) )
Applying Theorem 6.4 setting A=x ̂ and B=C=p ̂, the commutator can be expand leaving,
█(d/dt 〈x ̂ 〉=1/2imℏ 〈([x ̂,p ̂ ] p ̂+p ̂[x ̂,p ̂ ])〉.#(6.24) )
Utilizing the commutator result [x ̂,p ̂ ]=iℏ,
█(d/dt 〈x ̂ 〉=1/2imℏ 〈(iℏp ̂+p ̂iℏ)〉=1/2imℏ 〈2iℏp ̂ 〉,#(6.25) )
█(d/dt 〈x ̂ 〉=〈p ̂ 〉/m.#(6.26) )
This result can be compared with p=mv from classical mechanics. It is also possible to translate it into an expression involving the Hamiltonian, only if it is legal to take the derivative of the Hamiltonian operator with respect to another operator, namely p ̂ as shown,
█(d/dt 〈x ̂ 〉=〈p ̂ 〉/m=〈(∂H ̂)/(∂p ̂ )〉.#(6.27) )
This clearly shows a correspondence with one of Hamilton’s equations seen in (5.12),
█(d/dt 〈x ̂ 〉=〈(∂H ̂)/(∂p ̂ )〉 → x ̇_k=∂H/(∂p_k ).#(6.28) )
The similarities between the classical and quantum results here are rather striking. It is clear to see that the quantum methods using operators and commutators predict the same outcomes as classical mechanics for the time evolution of a system.
Example 6.7- We now follow a similar route as in [15] using A ̂=p ̂ the operator for momentum in the Ehrenfest theorem,
█(d/dt 〈p ̂ 〉=1/iℏ 〈[p ̂,H ̂ ]〉+〈(∂p ̂)/∂t〉.#(6.29) )
Again p ̂ has no time dependence so the second term disappears. Using the same Hamiltonian (6.20)
█(d/dt 〈p ̂ 〉=1/iℏ 〈[p ̂,p ̂^2/2m+V(x)]〉.#(6.30) )
Here p ̂ commutes with p ̂^2 and so we are left with [p ̂,V(x)],
█(d/dt 〈p ̂ 〉=1/iℏ 〈[p ̂,V(x)]〉=1/iℏ 〈-iℏ dV(x)/dx〉.#(6.31) )
By utilizing the result from (6.10) for the commutator. Some trivial simplification leaves,
█(d/dt 〈p ̂ 〉=-〈dV(x)/dx〉=〈F〉.#(6.32) )
In one dimension, we can see that the rate of change of the average momentum is equal to the average derivative of the potential V. Again, the behavior of the average Quantum variables corresponds with the classical expressions for these observables. In classical terms (6.32) reduces to F=dp/dt=-dV/dx . Here again we require the derivative of V to be defined and the derivative of the Hamiltonian to be legal, assuming these are satisfied we have,
█(d/dt 〈p ̂ 〉=-〈dV(x)/dx〉=-〈∂H/∂x〉.#(6.33) )
Again, one sees resemblance between this quantum result and the classical Hamilton’s equations (5.12),
█(d/dt 〈p ̂ 〉=-〈∂H/∂x〉 →〖 p ̇〗_k=-∂H/(∂x_k ).#(6.34) )
Again, the key difference between the classical and quantum versions of the above equations is that the quantum results show a relation between the average mean values expected to be taken by an observable. Whereas, the classical interpretation gives a definite value of the observable. It was this correspondence that influenced Heisenberg in his formulation of quantum mechanics. In general, the Ehrenfest theorem does not confirm that the expectation value of an operator follows a classical trajectory. However, for small wavelengths (i.e. Macroscopic particles) it will approximate the classical paths.
The correspondence principle states that in systems which can be considered classical the predictions made by quantum mechanics must correspond to those of classical mechanics. [12] Here we have seen how the Ehrenfest theorem obeys this principle and next we shall see how the harmonic oscillator also bares an interesting analogy.
7. Simple Harmonic Oscillator
Example 7.1- Lagrangian Harmonic Oscillator [6]
Consider a system containing the undamped harmonic oscillator in 3-D, with displacement coordinate x= (x,y,z), which is a generalised coordinate. We first form a Lagrangian relation for this system,
█(L(x,x ̇,t)=1/2 mx ̇^2-1/2 kx^2.#(7.1) )
Now, we consider the case of the 1-D harmonic oscillator (i.e. Constraining y and z to both be zero, x= (x,0,0)). [4] Leaving us to find the following equations,
█(∂L/(∂x ̇ )=mx ̇,∂L/∂x=-kx.#(7.2) )
Hence, our equations of motion for the system,
█(d/dt (∂L/(∂x ̇ ))-∂L/∂x=mx ̈+kx=0.#(7.3) )
All that is left is to rearrange this equation and to solve,
█(mx ̈=-kx ⟹Solutions of form x(t)=A cos(ωt)+B sin(ωt
),where ω≔√(k/m).#(7.4) )
Definition 7.2- Scaled quantum operators for position and momentum X ̂ and P ̂ can be defined as,
█(X ̂=√(mω/ℏ
) x ̂,P ̂=p ̂/√mℏω.#(7.5) )
Hence, lowering and raising operators a〖 and a〗^† can be defined in the following way,
█(a=1/√2 (X ̂+iP ̂ ) and a^†=1/√2 (X ̂-iP ̂ ) #(7.6) )
They have commutation relation,
█([a,a^† ]=1#(7.7) )
The lowering and raising operators are also known as ladder operators, which are useful when analysing the quantum harmonic oscillator. [13]
Remark 7.3- The theory of quantum mechanics makes predictions using probabilities for the result of a measurement of an observable A ̂. The probabilities are found by obtaining the real eigenvalues a_i of A ̂ and using the relation stated in the postulates,
█(Prob(a_i )=|⟨a_i │ψ⟩|^2.#(7.8) )
Example 7.4- Quantum Harmonic Oscillator [13]
We start with our scaled operators of position and momentum,
█(X ̂=√(mω/ℏ) x ̂,P ̂=p ̂/√mℏω.#(7.9) )
For the quantum harmonic oscillator, we need a Hamiltonian operator based on the classical simple harmonic oscillator. Replacing observables x and p with operators we have
█(H ̂=p ̂^2/2m+1/2 mω^2 x ̂^2=1/2 ℏω(X ̂^2+P ̂^2 ).#(7.10) )
We use lowering and raising operators a and a^† defined in (7.6) in order to find the wave function for the simple harmonic oscillator. We have scaled operators of position and momentum as in (7.9), so we can write P ̂ in terms of our X,
█(p ̂=-iℏ ∂/∂x ⟹ P ̂=p ̂/√mℏω=-i ∂/∂X.#(7.11) )
Lowering operator a can act in our X-space on ground state ket |├ n_0 ⟩ (the state which cannot be lowered anymore by the lowering operator). Such that,
█(⟨X│a│n_0 ⟩=1/√2 (⟨X│X ̂│n_0 ⟩+i⟨X│P ̂│n_0 ⟩)=0,#(7.12) )
as we cannot lower past the ground state. Apply the definition of the expectation values,
█(⟨X│a│n_0 ⟩=1/√2 (∫▒〖⟨X│X ̂│X^’ ⟩⟨X^’│n_0 ⟩ dX^’ 〗 +i∫▒〖⟨X│P ̂│X^’ ⟩⟨X^’│n_0 ⟩ dX^’ 〗).#(7.13) )
Evaluating the two terms inside the bracket we see,
█(⟨X│X ̂│n_0 ⟩=∫▒〖⟨X│X ̂│X^’ ⟩⟨X^’│n_0 ⟩ dX^’ 〗=∫▒〖δ(X-X^’ ) X^’ 〗 ψ_0 (X^’ ) dX^’= Xψ_0 (X),#(7.14) )
█(⟨X│P ̂│n_0 ⟩=∫▒〖⟨X│P ̂│X^’ ⟩⟨X^’│n_0 ⟩ dX^’ 〗=∫▒〖-iδ(X-X^’ ) ∂/(∂X^’ ) ψ_0 (X^’ ) dX^’ 〗=-i (∂ψ_0 (X))/∂X.#(7.15) )
So, we have equation (7.13) rewritten as,
█(1/√2 (Xψ_0 (X)+(∂ψ_0 (X))/∂X)=0.#(7.16) )
Giving us solution the solution for our ground state wave function,
█(ψ_0 (x)=Ae^((-mωx^2)/2ℏ) and A=(mω/πℏ)^(1/4).#(7.17) )
Now we have our ground state we can apply raising operator a^† to |├ n_0 ⟩ and using a similar approach to above,
█(ψ_1 (x)=⟨x│a^†│n_0 ⟩=1/√2 (mω/πℏ)^(1/4) (X-∂/∂X) e^((-mωx^2)/2ℏ).#(7.18) )
By repeating this process, at the end of the story we find a generalised form of the normalised wave function,
█(ψ_n (x)=1/√(2^n n!) (mω/πℏ)^(1/4) H_n (X) e^((-mωx^2)/2ℏ),#(7.19) )
where H_n are Hermite polynomials.
Figure 7.1: [16]
Left: Shows |〖ψ_0 (x)|〗^2 against the classical probability density.
Right: Shows |〖ψ_10 (x)|〗^2against the classical probability density.
We can compare the probability density function of the classical approach with the quantum ground state |〖ψ_0 (x)|〗^2. It is clear to see that the classical mechanics has a minimum at x=0, where it has maximum kinetic energy, whereas for quantum mechanics peaks at x=0 for the ground state. However, as n increases the quantum wave functions begin to represent a similar distribution to that of classical mechanics as shown in figure 8.2. For a very large n with macroscopic energies, the classical and quantum curves are indistinguishable, due to limitations of experimental resolution.
Conclusion
In this report, we have defined the basics Lagrangian mechanics including the principle of least action and derived the Euler-Lagrange equation. From this equation, we were able to define the generalised momenta and conservation laws of a system. The Lagrangian formulation is useful compared to Newtonian mechanics as it uses a set of generalised coordinates which can take into account any constraints put on the system. We then saw how Hamiltonian mechanics can be formulated from the Lagrangian approach and even though there is no new physics present it can have numerous benefits in other branches of mechanics. Next, we introduced quantum mechanics and the Ehrenfest theorem. We saw how this theorem in comparison with a classical expression can relate the Poisson bracket and the quantum commutator with a simple factor of iℏ. Then, we applied the theorem to operators for position and momentum and again it was clear that there was an analogy between these quantum results and those found in Hamiltonian mechanics. Finally, using a system containing a simple harmonic oscillator we were able to directly calculate solutions using both classical and quantum methods. From these results, we saw that as n increases the quantum wave functions begin to represent a similar distribution to that of classical mechanics.
In further study, one could explore the Hamilton-Jacobi equations as a follow on from Hamiltonian mechanics, these, as well as a simple action containing the Lagrangian can be used in the derivation of quantum mechanics. [17] [18] Furthermore, one could study Bohr’s correspondence principle where we take large values of n in a quantum system and see how that compares to classical results.
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