Derive Time Independent SCHRODINGER's EQUATION from Time Dependent one
Summary
TLDRIn this educational video, the presenter demonstrates how to transform the time-dependent Schrödinger equation into its time-independent form using the separation of variables method. This technique is crucial in quantum mechanics for analyzing the behavior of particles under potential fields that are time-independent. The video elucidates the process of reducing a complex partial differential equation to a simpler ordinary differential equation, which is essential for solving various quantum mechanical problems and understanding properties like energy levels and wavefunctions.
Takeaways
- 🔬 The video demonstrates the process of converting the time-dependent Schrödinger equation into its time-independent form using the separation of variables method.
- 🌌 In quantum mechanics, the Schrödinger equation is fundamental for studying the evolution of quantum systems, much like Newton's second law in classical mechanics.
- 📉 To use the separation of variables, the potential in the quantum system must be independent of time, allowing the wave function to be expressed as a product of spatial and temporal functions.
- ⏱️ The time-dependent Schrödinger equation is a partial differential equation involving both space and time derivatives, which can be simplified under the right conditions.
- 🧮 The reduction to an ordinary differential equation is achieved by assuming the wave function can be separated into functions of space and time, leading to two separate equations.
- 🔄 The separation constant, denoted as 'G', equates the spatial and temporal equations, suggesting that the energy of the system is related to the frequency of the time-dependent part of the wave function.
- 🌊 The time-independent part of the wave function is represented by a function of space only, which is crucial for solving the time-independent Schrödinger equation.
- 🌟 The solution to the time-independent equation provides eigenfunctions, which, when multiplied by the time-dependent part, give the complete wave function of the system.
- 📊 The probability density, derived from the wave function, remains constant over time for systems where the potential is time-independent, indicating stationary states.
- 🔍 The video concludes by emphasizing that the time-independent Schrödinger equation is essential for further analysis in quantum mechanical problems where the potential does not vary with time.
Q & A
What is the main topic of the video?
-The main topic of the video is the reduction of the time-dependent Schrödinger equation to its time-independent form using the separation of variables method.
Why is it important to study the trajectory of a quantum mechanical particle?
-Studying the trajectory of a quantum mechanical particle is important because it allows us to understand the evolution of the system, and from the solution of the Schrödinger equation, we can derive various physical quantities like position, momentum, velocity, and acceleration.
What is the fundamental equation in quantum mechanics that is discussed in the video?
-The fundamental equation in quantum mechanics discussed in the video is the Schrödinger equation.
How does the separation of variables method simplify the Schrödinger equation?
-The separation of variables method simplifies the Schrödinger equation by allowing the wave function to be expressed as a product of two separate functions, one dependent on space and the other on time, which reduces the partial differential equation to an ordinary differential equation.
Under what condition can the wave function be separated into space and time functions?
-The wave function can be separated into space and time functions when the potential field experienced by the particle is independent of time and only depends on the spatial variable.
What is the significance of the separation constant (G) in the context of the Schrödinger equation?
-The separation constant (G) in the Schrödinger equation is significant because it represents the energy of the particle in the quantum mechanical system, linking the frequency of the wave function to the energy of the particle through Planck's constant.
How is the time-independent Schrödinger equation derived from the time-dependent one?
-The time-independent Schrödinger equation is derived by separating the wave function into space and time components and setting the separated spatial and temporal parts equal to a constant, which leads to two ordinary differential equations, one for the spatial part and one for the temporal part.
What is the physical interpretation of the solution to the time-independent Schrödinger equation?
-The solution to the time-independent Schrödinger equation, often referred to as the eigenfunction, represents the spatial part of the wave function and is associated with the stationary states of the quantum system, where the probability density of the particle is constant over time.
Why are the solutions to the time-independent Schrödinger equation also called eigenfunctions?
-The solutions to the time-independent Schrödinger equation are called eigenfunctions because they correspond to the eigenvalues of the Hamiltonian operator, which in this context represents the total energy of the system.
How does the video explain the relationship between the frequency of a particle's wave and its energy?
-The video explains the relationship between the frequency of a particle's wave and its energy by referencing Planck's and Einstein's postulates, which state that the frequency (ν) is related to the energy (E) by the equation E = hν, where h is Planck's constant.
What is the final form of the wave function solution according to the video?
-According to the video, the final form of the wave function solution is a product of the eigenfunction, which is a solution of the time-independent Schrödinger equation, and the time-dependent part, which is an exponential function representing the oscillatory behavior of the system.
Outlines
🔬 Introduction to Time-Dependent Schrödinger Equation
The video begins with an introduction to the process of simplifying the time-dependent Schrödinger equation into its time-independent form using the method of separation of variables. The presenter explains that in quantum mechanics, the Schrödinger equation is fundamental for studying the evolution of quantum systems. The trajectory of a quantum particle can be analyzed by solving this equation, which provides insights into various physical quantities like position, momentum, and velocity. The video aims to demonstrate how to reduce the complexity of the time-dependent equation by separating variables, a method applicable when the potential field is time-independent.
📐 Separation of Variables Method
This section delves into the mathematical process of using the separation of variables method to transform the time-dependent Schrödinger equation into an ordinary differential equation. The presenter explains that by assuming the wave function can be expressed as a product of two separate functions—one dependent on space and the other on time—the partial differential equation can be simplified. The conditions under which this method is applicable are discussed, specifically when the potential field does not vary with time. The video illustrates how to rearrange the equation and separate it into two ordinary differential equations, each dependent on a single variable.
🌌 Time-Independent Schrödinger Equation and Its Solutions
The presenter continues by solving the time-independent Schrödinger equation, which is derived from the separation of variables. The solution involves finding the eigenfunctions and eigenvalues, which are crucial for understanding the quantum states of a system. The video explains how the time-dependent part of the wave function can be represented as an exponential function of time, which includes an imaginary unit. The solutions are then decomposed into cosine and sine functions, revealing the oscillatory nature of the quantum states. The relationship between the frequency of these oscillations and the energy of the particle is discussed, linking to Planck's and Einstein's theories.
🌟 Conclusion: Stationary States and Probability Density
The final part of the video concludes with the derivation of the time-independent Schrödinger equation and its implications for quantum mechanical systems. The presenter demonstrates that the solution to the equation can be expressed as a product of spatial and temporal components, with the spatial part being the eigenfunction. It is explained that these solutions represent stationary states, where the probability density of the particle remains constant over time. The video wraps up by emphasizing the importance of solving the time-independent equation for various quantum mechanical problems and how it leads to a constant probability distribution, indicative of the system's stationary nature.
Mindmap
Keywords
💡Schrodinger Equation
💡Separation of Variables
💡Wave Function
💡Eigenfunction
💡Quantum Mechanical System
💡Potential Field
💡Time-Independent Schrodinger Equation
💡Stationary States
💡Probability Density
💡Planck's Constant
Highlights
Introduction to reducing the time-dependent Schrödinger equation to its time-independent form using the separation of variables method.
Fundamental importance of Schrödinger's equation in quantum mechanics for studying the evolution of quantum mechanical particles.
The trajectory of a particle in classical mechanics is studied using Newton's second law, analogous to Schrödinger's equation in quantum mechanics.
The time-dependent Schrödinger equation is a partial differential equation involving space and time variables.
Condition for reducing the partial differential equation to an ordinary differential equation through the separation of variables.
The potential field must be independent of time for the separation of variables method to be applicable.
The wave function can be expressed as a product of space and time functions under certain conditions.
Derivation of the time-independent Schrödinger equation from the time-dependent form.
The separation constant 'G' is introduced, equating terms involving different variables.
Solving the time-independent Schrödinger equation provides the eigenfunction, a function of space only.
The time-dependent part of the wave function is solved as an exponential function of time.
The solution involves an imaginary number and Planck's constant, leading to an oscillatory function.
The frequency of the oscillatory function is related to the energy of the particle through Planck-Einstein relation.
The energy of the particle in a quantum mechanical system is represented by the separation constant 'G'.
The final wave function solution is a product of the eigenfunction and the time-dependent function.
Eigenfunctions represent stationary states where the probability density is constant with respect to time.
The probability density of the particle is independent of time, proving the stationary nature of the states.
Conclusion on how the time-dependent Schrödinger equation is reduced to a time-independent form using separation of variables.
Transcripts
so in today's video I want to show how
to reduce a time-dependent square inches
equation form into its type independent
form by using the separation of
variables method so in quantum mechanics
one of the most fundamental equations is
disk orange's equation now what happens
is that if you want to study some kind
of quantum mechanical system you want to
study the evolution of a quantum
mechanical particle you want to study
the trajectory of a particle the most
the first thing that you need to do is
solve this Cottagers equation and from
the solution you can study you can get
idea about many different physical
quantities like position momentum
velocity acceleration and all these
different things associated with that
particular system for example in
classical mechanics what you basically
do if you want to study the trajectory
of a particle is that you basically have
the Newton's second law you if you know
all the forces and you know the position
and the velocity of a particle at a
given instant in time they then you
solve the Newton's second law from that
you can find out the position with
respective time time derivative of the
position gives you an idea about the
velocity the time derivative of the
velocity gives you an idea about the
acceleration so in the same way in
quantum mechanics if you want to study a
particular system or a particle which is
bound by some kind of a force field or a
potential then you basically solve a
scourges equation and from in that
solution different kinds of information
is contained in it and how you proceed
from there on is a completely different
matter but in this video I what I want
to show is how to reduce the time
dependent Schrodinger equation to its
time independent form so the most basic
time dependence cod inches equation is
of this form
[Applause]
so as you can see in the time dependence
car in jersey collision the
wavefunctions solution sigh here is
actually a function of both space and
time looming and the potential itself is
also a function of X and T now this this
as you can see that this is a partial
differential equation which contains two
partials arrive at it so one is the
second order partial derivative of X and
the other is a first-order partial
derivative of time and I can reduce this
partial differential equation into an
ordinary differential equation if if
certain conditions are met so what
conditions on top keyboard so for
example if this particular wave function
which is both a function of X and T can
be written as product of two separate
functions one is a function of space and
the other is a function of time in that
case I can reduce this expression to an
ordinary differential equation now what
happens is that this kind of condition
is true only in those cases whenever the
particle is experiencing some kind of a
potential field where the potential
field is independent of time so
basically I have a potential field which
is independent of time and it is only
dependent on X here so for example
gravitation field electrostatic field so
a particle is experience it feels like
that which is independent of time in
those cases the wave function solution
is usually can be written as a product
of two separate functions both being
functions of independent separate
variables now if this this condition is
being met so let's say this is condition
number one and this is condition number
two so if I use so obviously these are
both related to each other so this is
known as separation of variables method
so separation of variables okay so if if
I have this particular condition I can
use this condition to reduce this
expression into a much simpler form
which can be solved for different kinds
of
quantum mechanical problems so now let's
replace equation number one in this
particular storages equation so if I do
that my equation becomes minus H cross
square by 2 m del square by Del X square
this this simple sy X plus Phi T plus V
X is equal to IH cross del Y del T off
[Applause]
now in the first term here now this is a
second order different partial
derivative of this particular product
but this product contains two separate
functions one is the function of space
there is a function of time domain since
X and T both are independent variables
so I can take this term outside the
derivative expression so this becomes
minus H cross square by 2 m Phi G del
square X by Del X square plus V X Phi X
Phi T is equal to IH cross similarly in
this term I can take the function of X
variable outside the partial derivative
of time in this case so IH cross Phi X X
my del T now one thing to notice is that
the moment I take these terms out side
the derivative expression this suddenly
becomes an ordinary derivative right
this becomes an ordinary gravity does
not remain a partial derivative anymore
because the function here is a function
of only one variable so it becomes the
ordinary derivative now to simplify it I
what I do is so this is Phi T I take
this out minus H cross square by 2 m
okay now taking these two expressions on
the other side interchanging them I get
so or so I get 1 by Phi X minus H cross
square by 2 m now as you can see this
differential equation has been reduced
to an ordinary differential equation in
which on the left hand side all these
terms all both of these two terms are
basically functions of X there is no
dependence on the variable of time T and
on the right hand side this term here is
as dependent is a function of time there
is no dependence of X now X M T these
are both independent variables and on
the left hand side if you have some kind
of a function of X and on the right hand
side if you have some kind of a function
of T and both these two expressions are
equal to each other since there is no
dependence of since both X and T are
independent of each other then the only
conclusion that you can derive from this
kind of an equation is that they are
both equal to some kind of a constant
and let's denote this kind of a constant
value to be let's say something like G
okay G is a separation constant so we
call G as the separation constant G okay
now if I if I if I do this then I can I
can separate both these two equations in
separate this entire equation do two
different equations so I can write that
minus H cross square by 2 m d square
side by DX square plus v sy is equal to
G I let's say this is point number three
and we also have IH cross D Phi by DT is
equal to G Phi T yes so now let's look
at the fourth equation let's look at the
fourth equation
in the fourth equation you have AI H
cross the Phi by DT is equal to G Phi or
D Phi by DT is equal to 1 by h plus G
Phi which can be written as if you take
the energy number I upwards this is
going to become minus I by H cross G Phi
or this is written as D Phi by DT is
equal to I by H cross hi gh plus fight
right so I is the imaginary number H
cross is Appliance constant divided by 2
pi G is the separation constant which is
equal to both these two different
expressions okay
now you solve this equation is very
simple so if you if you if you have some
equation of the form let's say dy by DT
is equal to let's say some kind of a
constant alpha multiplied by Y okay
because this entire equation is a this
form then the solution of this kind of
an equation is written as y is equal to
Y is equal to e to the power alpha T
this is a solution of an equation like
that y you can check that if I find the
first order derivative of D Y with
respect to time then dy by DT is equal
to so D by DT of e to the power alpha
which is basically the alpha is a
constant comes outside alpha e to the
power alpha T so either alpha is equal
to Y so this can be written as L so Y
alpha Y so dy by DT is equal to alpha y
is the different first order
differential equation whose solution is
y is equal to EI fod okay
so now I will use this conclusion here
that if we have this constant I G by H
cross is equal to some kind of a
constant alpha
IgH crosses some prolific constant he'll
find that case the solution Phi T I'm
sorry the - is here there's a minus sign
so - this thing okay so Phi T is equal
to Phi T is equal to e to the power
alpha T which is equal to e to the power
minus i g by h cross so this is the
solution of equation number four okay
this is the solution of equation number
four so equation number four has a
solution of Phi T is equal to e to the
power minus i g pi h cross t okay this
is a solution now let's look at this
particular solution the solution is
basically exponential to the power
something multiplied by time
now what and this sum this constant also
includes an Imagineer number now if you
have you have expressions of the form e
to the power i Tita then they are
basically written as cos theta plus I
sine theta right and if you have
expressions of re to power minus I theta
this can be written as cos theta minus I
theta so if I if I if I if I also
decomposed this particular expression in
the terms of cosines and sines then
basically I can write as e to the power
minus i g by H cross T is equal to
course cost
G by H cross t minus I sine G to the
power H cross okay where H cross is
nothing but H by 2 pi so this is
basically cause G by 2 pi G by h t minus
I sine 2 pi G bar H okay now as you can
see here
this is some kind of an oscillatory
function this is some kind of an
oscillator and the oscillator dependence
has a frequency of is equal to cos 2 pi
mu t minus I sine 2 pi right where the
frequency nu is is equal to G by H okay
because these are not just mathematical
functions they are supposed to represent
a some kind of particle which is
experience experiencing some kind of a
potential field now if if we have a time
dependence which is oscillatory in
nature then that oscillation of the that
time dependent function has an
expression which is equal to G by H but
we already know as given by as given by
Einstein and Planck's equation is that
the frequency of any kind of a wave
associated with some kind of a particle
has a relationship with Planck's
constant and energy which is expression
is given by e is equal to H nu so this
comes from Planck Einsteins postulate so
this expression was most initially given
by Max Planck when he was trying to
explain the phenomena of blackbody
radiation and later on it was also
followed by I'm trying to explain the
photoelectric effect so we already know
that whenever we have some kind of a
particle which has a wave associated
with it and the frequency of the wave is
related to its energy by this kind of an
expression so so by looking at this
similarity of this expression we can
conclude that this or G is equal to H u
in this case G is nothing but the energy
of that particle
stuck in that kanafeh given quantum
mechanical system so G is actually equal
to the energy by comparing the Max
Planck I'm Stein wash rate of relating
frequency with that of the energy and
the conclusion that we came up both from
here so G is nothing but the energy of
the particle stuck in that kind of a
quantum mechanical system so if G is
equal to e in that case so J is equal to
EE then equation number four equation
number four gives us the solution Phi T
is equal to e to the power minus i ii by
H cross P so this is the function which
shows you the dependence on time and
equation number three takes the form of
minus H cross square by 2m d square side
by DX square yes v sy is equal to e sy
so this here this equation here is the
time independent Schrodinger equation
this is the time independent scrod
inches equation which can be further
used in whatever problem we are studying
to to to know about the solution so
these the sy here is known as the
eigenfunction and sy is nothing but a
function of X now for all kinds of cases
of quantum mechanical systems where we
can use separation of variable methods
if the potential field is independent of
time in all of those cases the time
dependence can be written in this form
and then we have the space dependence
given by the time independent
Schrodinger equation so the next step is
to obviously solve this equation
depending upon what kind of problem we
have at hand so the final solution so
the final wave function solution
therefore can be represented as wave
function solution can be represented as
a product of this eigenfunction side and
T here so this can be written as
[Applause]
so the solution of the score dangers
equation has a wave function which can
be written in this form and this is the
eigen function which is a solution of
the time-independent orangist equation
now this is also known as eigen function
or sometimes also known as stationary
States because this is independent of
the time domain and if you solve this
time independent religious equation for
some particular quantum mechanical
problems then you will find that the
probability density of the particle for
those kind of cases are always constant
with respect to time so the there is the
the so if this is the case then the
probability density of the particle
which is a wave function solution that
looks like this is independent of
independent of time how how can you
prove that so for example if you have if
you want to find the probability
probability probability distribution of
a particle you basically do si star sy
DX right so if you do this so this this
exponential terms gets cancelled out and
you're left to it this particular term
only which is independent of x the
probability distribution is independent
of time and therefore this is also you
end up getting stationary States so this
is how you can reduce the time
dependents what is equation to a time
independent form by using the separation
of variables method thank you
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