Uncertainty, Probability, & Normality Flashcards
It is described as a better theory to describe the behavior of matter at the atomic level.
Quantum mechanics
It provides a theoretical background that makes predictions that agree with experiment.
Quantum mechanics
Quantum mechanics is based on several statements called __________.
postulates
These postulates are ________, not proven
assumed
Why do we believe these postulates even if not proven?
the statements based on these assumptions lead to predictions about atoms and molecules that agree with our observations.
The wavefunction contains ___________.
within it all possible information that can be known about a system.
Wavefunctions are not _________ mathematical functions, but must satisfy certain simple conditions
arbitrary
The most important condition is that the wavefunction must satisfy the _________________
time-dependent Schrödinger equation.
Quantum mechanics predicts values that __________________________________.
agree with experimentally determined measurements
The state of a system can be described by
an expression called a __________________.
wavefunction
What are the limitations of a wavefunction?
single valued, continuous, bounded, differentiable/integrable
Functions that meet all these criteria are considered _______________________.
acceptable wavefunctions
All possible information about the various observable properties of a system must be _______________________.
derived from the wavefunction
Give examples of an observable.
mass, volume, position, momentum, and energy
In order to determine the value of an observable, you have ______________.
to perform some mathematical operation on a wavefunction
When an operator acts on a function and produces the original function multiplied by any constant the equation is referred to as an ____________.
eigenvalue equation
Another postulate of quantum mechanics states that for every ______________ of interest, there is a corresponding ___________.
physical observable
operator
The only values of the observable that will
be obtained in a single measurement must be _____________________-.
eigenvalues of the eigenvalue equation constructed from the operator and the wavefunction
The eigenvalue equation is constructed from the ____________________.
the operator and the wavefunction
the eigenvalue equations that we will consider have ______________ as values of eigenvalues
real numbers
Heisenberg’s full name
Werner Karl Heisenberg
__________________ completely changed the way science understands the limitations in the ability to measure nature.
Heisenberg’s uncertainty
principle
Hermitian operators are operators that always have ____________________ as eigenvalues .
real (nonimaginary) numbers
All operators that yield quantum mechanical observables are Hermitian operators, ________________________.
because in order to be observed a quantity must be real
Hermitian operators are named after _________________, a nineteenth century French mathematician
Charles Hermite
When did Werner Karl Heisenberg announce his uncertainty principle?
1927
The uncertainty principle states that there are ultimate _________ to how exact certain measurements can be
limits
The de Broglie wavelength is related to a _______________.
momentum
The uncertainty principle deals only with certain observables that might be measured simultaneously. Two of these observables are__________________.
position and momentum
The uncertainty principle puts a ____________ on the uncertainty.
lower bound
Full name of German scientist Born
Max Born
dT is called?
infinitesimal of integration
The Born interpretation also requires that a probability be evaluated over a ____________, not a specific point, in space.
definite region
Instead of wavefunction giving the exact location of an electron, it will provide only the ________________________.
probability of the location of an electron
If the probability for a particle having wavefunction were evaluated over the _____________________ in which the particle exists, then the probability should be equal to 1, or 100%
entire space
In order to achieve 100% probability, a wavefunction must be ______________.
normalized
The ___________________ does not affect the shape of the function.
normalization constant
STATE WHICH POSTULATE
The state of a system of particles is given by a wavefunction.
Postulate 1
STATE WHICH POSTULATE
For every physical observable or variable there exist a corresponding Hermitian operator.
Postulate 2
STATE WHICH POSTULATE
The only values of observables that can be obtained in a single measurement are the eigenvalues of the eigenvalue equation.
Postulate 3
STATE WHICH POSTULATE
Wavefunction must satisfy the time-dependent Schrodinger equation.
Postulate 4
The Heisenberg’s uncertainty principle states that it is__________.
impossible for us to know simultaneously both the exact momentum of an electron and its exact location in space.
It is very difficult to establish absolutely that a particular electron is in a particular place at a particular time. Rather, over a long period of time, the electron has a certain probability of being in a__________.
certain region
If the probability for a particle having wavefunction were evaluated over the entire space in which the particle exists, then the probability should be equal to ________________
1 or 100%.
The integral’s _______ would be modified to represent the ________ of the space a particle
inhabits
limits
limits
wavefunctions must be multiplied by some constant, called the _____________, so that the area under the curve of Ψ* Ψ is equal to 1.
normalization constant
According to the Born interpretation of Ψ, normalization also guarantees that the probability of a particle existing in ______________ is 100%.
all space
It only imposes a scaling factor on the amplitude—a very convenient scaling factor.
normalization constant
___________________ deals with the most important observable: energy
The Schrödinger Equation
______________ proposed an expression of quantum mechanics that was different from but equivalent to Heisenberg’s.
Erwin Schrödinger
The Schrödinger equation is based on the _____________________, because these equations naturally produce the total energy of the system:
Hamiltonian function
If is a ________________ (that is, if its probability distribution Ψ does not depend on time), it must also satisfy the Schrödinger equation
stationary state
When applied to ideal and even real systems, it yields the values for the ____________________________.
energies of the systems
The behavior of electrons is described by a ___________________.
wavefunction
The appropriate operator for predicting the energy of the electron is the ___________________.
Hamiltonian operator
The ____________ energy part of the Hamiltonian has a similar form for all systems. However, the ___________ energy operator V^ depends on the system of interest
kinetic
potential
Solutions that have a specific mathematical form, either a number or an expression.
analytic solutions
This is the first system for which there is an analytic solution is a particle of matter
stuck in a one-dimensional “prison” whose walls are infinitely high barriers. What do you call this system.
the particle-in-a-box
In the system particle-in-a-box, the potential energy inside the box itself is ___________. While outside it is _________.
zero
infinity
This requirement, that the wavefunction must be a certain value at the boundaries of the system, is called a _______________.
boundary condition
It is referred to as the three-dimensional operator.
Laplacian operator
As the mass of the object increases, the momentum increases, then the uncertainty __________.
decreases
