Download Quantum Mechanics Bohr`s model: - one of the first ones to use idea

Quantum Mechanics
Bohr's model:
- one of the first ones to use idea of matter waves to solve a problem
- gives good explanation of spectrum of single electron atoms, like
hydrogen
BUT:
- does not explain even simple many electron systems, like Helium
- does not explain intensity of spectral lines
- does not help to explain structure of atoms, their binding to other
atoms, or any physical of chemical properties of interacting atoms
In 1925-26:
- another more general approach to explain properties of microscopic
system such as atoms is developed by Erwin Schrödinger, Werner
Heisenberg, Max Born, Paul Dirac and others
- it is called quantum mechanics
by 1930s:
- a vast number of experiments in physics and chemistry is
successfully explained using this new theory
- up to now the theory of quantum mechanics has been proven to be
correct by many experiments
- quantum mechanics is a very successful theory
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Goal:
- develop a theory that describes the wave properties of particles correctly
steps:
- use a complex valued wave function ψ to describe the particle
- the modulus squared of ψ evaluated at a specific coordinate and time is
proportional to the probability density P to find a particle at that
coordinate and at that time
- ψ∗ is the complex conjugate wave function to ψ
properties of ψ:
- the wave function is normalized
- if ψ describes the particle and |ψ|2 its probability density then the total
probability of finding the particle somewhere has to be unity (=1) for the
wave function to be a valid description of the particle
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properties of ψ:
- single valued and continuous
- single valued and continuous derivatives
probability
- to find a particle in a region of space (for simplicity in one
dimension):
Schrödinger equation:
- fundamental equation of quantum mechanics
- the wave equation for matter waves
- is postulated and cannot be derived (like other fundamental
equations, laws of thermodynamics, Newton's equation of
motion, …)
exercise:
- work out some examples of wave equations to remember some
of their properties and get some familiarity working with
these
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time-dependent Schrödinger equation:
wave function:
- for a free particle described by a plane wave along the xdirection with velocity v = λ ν
- express ν and λ in terms of energy and momentum
- thus
- with the reduced Planck constant
- This wave function describes a free particle. How to find the
wave function describing a particle in the presence of external
constraints, e.g. a potential resulting in some force acting on the
particle?
goal:
- Find a fundamental differential equation the solution to which
is the wave function describing the properties of the particle.
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Schrödinger equation:
- a strict derivation from first principles is not possible
- let's develop some intuition trying to set up a wave
equation for ψ by differentiating it with respect to x
- and with respect to t
- for v < c the total energy of a particle is given by
- multiplying with ψ
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- substitute expressions for E and p
- time-dependent Schrödinger equation (1D)
- in three dimensions (3D)
- the potential function U contains all the external forces
acting on the particle
- solving the Schrödinger equation for ψ the probability
density for the particle can be found for any coordinate and
time
- again: Schrödinger's equation is a postulate, there is no
formal justification in the considerations presented above
- but the physical world as probed in exquisite detail in
experiments is very well described by this equation
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Linearity:
- the Schrödinger equation is linear in the wave function ψ
- it contains no terms that involve higher powers of ψ or it's derivatives
Superposition:
- as a result if ψ1 and ψ2 are solutions of a S.E. the superposition of
both is also a solution
- a1 and a2 are complex coefficients
- the superposition principle applies to matter waves in similar ways as
to any other waves
- the superposition principle gives rise to interference effects in
quantum mechanics
single solution terms
two-solution terms
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Expectation values:
- all information about the particle is contained in the wave
function ψ
- we know already how to calculate the probability to find a
particle at a certain coordinate and time
- to extract other experimentally accessible quantities
expectation values are calculated
position expectation values:
- classical center of mass for a collection of N particles with
Ni particles at position xi
- for a single particle with probability pi at position xi
- in the continuous limit
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general:
- expectation values of quantity G(x) (that is not an operator!)
- What about the expectation values of the momentum p and the energy E?
- differentiate the free particle wave function ψ with respect to x and t
- correspondence
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Operators:
- momentum operator
- energy operator
- total energy operator:
- multiply with ψ to get back to the Schrödinger equation
- expectation values of an operator O
- for example for the momentum
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example:
- expectation value of the momentum p of a free particle
time independent (steady state) Schrödinger equation
- in many problems the potential U in the Schrödinger equation does not
depend on time explicitly
- in this case the Schrödinger equation can be simplified
- note for the free particle wave function
- for all time-independent problems the wave function can be separated
into these two parts
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time-independent (steady-state) Schrödinger equation:
- in three dimensions
- this steady-state Schrödinger equation may have one or more solutions
ψi
- the energy Ei corresponding to the different solutions ψi naturally
explain the energy quantization observed in any stable timeindependent system
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Eigenfunctions and eigenvalues of operators:
- consider an operator O
- if this equation holds ψn is an eigenfunction of the operator O with
eigenvalue on
- the equation above is also called an eigenvalue equation
measurement postulate:
- if a physical observable O corresponding to the operator O is measured
the only possible measurement outcomes are any of the on
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Hamilton operator:
- the total energy operator E of a system can be written in
terms of the momentum and potential energy operators p, U
in
- with the Hamilton operator H
- the Schrödinger equation thus is a eingenvalue equation for
the Hamiltonian H
- the En are the eigenenergies of a system described by the
Hamiltonian H, they are corresponding to the quantized
energies of the system
operators:
position
momentum
potential energy kinetic energy
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total energy
Particle in a box:
- a single particle trapped in a box with hard walls
- one of the simplest possible examples of solving a Schrödinger
equation
- let's treat the box as one-dimensional (1D)
- the particle is confined along the x-direction at x = 0 and at
x=L
- in the region 0 < x < L the potential is U = 0, elsewhere
U=∞
- plot of potential U
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Particle in a box:
- task: find the wave function ψ describing the particle in the
box by solving the Schrödinger equation
- 2nd order ordinary differential equation for ψ
- general solution
- boundary condition ψ = 0 for x < 0 and x > L
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boundary conditions:
- continued
- resulting allowed particle energies En
- these are the allowed energy levels of a particle in a 1D box
wave function:
- the corresponding (not yet normalized) wave functions
- normalization condition
normalized wave function:
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plots for different n:
- wave functions ψn
note:
- probability densities |ψn|2
- the probability densities vary a lot with n
- n = 1: largest probability in the center (x = L/2)
- n = 2: zero probability in the center
- classically probability is expected to be the same everywhere
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expectation values:
- position <x>
- momentum <p>
- the expectation value of the momentum vanishes! Why is that?
- expectation for momentum from energy E = p2/2m
- average momentum
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momentum eigenfunction:
- momentum eigenvalue equation
- energy eigenfunctions are not momentum
eigenfunctions
- momentum eigenfunctions
momentum eigenvalues
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Finite Potential Well:
- in a 1D potential well where U = 0 for 0 < x < L and U
= U0 < ∞ the particle can penetrate into the wall
- all natural potential wells have finite depth
- consider a particle with energy E < U0 less than the depth
of the potential
- three regions in the problem
- region II: U = 0
- region I and III: U0 > E
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general solution:
- in regions I and III
- otherwise ψ would diverge
- in region II
- with boundary conditions
continuity of
wave function
continuity of
momentum
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finite potential well
wave functions:
some notes:
probability densities:
- the particle is trapped forever
- how does the particles ever get into the potential well?
- how does it leave?
- physical examples of a particle in a box
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