Confused [Archive] - Physics Forums

davidelkins

Aug12-04, 08:29 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\n\nHello, my name is David Elkins. I have a question for quantum\nmechanics buffs out there. I am trying to learn the mathamatics behind\nQM and am (not surprisingly) having some difficulties. One of the\ndifficulties is my confusion over what are vectors and vector spaces\nin QM. In classical physics a vector is defined as a line segment with\nmagnitude and direction. In QM they use the term vector alot, but its\ndefinition is somewhat different.\n1. Do vectors in QM have directionality?\n2. Do they have magnitude?\n3. Can they be represented as a line segment?\n4. What is vectorlike about them?\n5. Or is attempting to understand QM vectors classically a hopeless\nendeavor?\n\nLastly, on the side, can anyone give as concrete as possible\ndescription of bra-ket notation? That\'s confusing the heck out of me.\n\nThankx, David,\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hello, my name is David Elkins. I have a question for quantum
mechanics buffs out there. I am trying to learn the mathamatics behind
QM and am (not surprisingly) having some difficulties. One of the
difficulties is my confusion over what are vectors and vector spaces
in QM. In classical physics a vector is defined as a line segment with
magnitude and direction. In QM they use the term vector alot, but its
definition is somewhat different.
1. Do vectors in QM have directionality?
2. Do they have magnitude?
3. Can they be represented as a line segment?
4. What is vectorlike about them?
5. Or is attempting to understand QM vectors classically a hopeless
endeavor?

Lastly, on the side, can anyone give as concrete as possible
description of bra-ket notation? That's confusing the heck out of me.

Thankx, David,

Maurice Barnhill

Aug12-04, 10:26 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>davidelkins wrote:\n> Hello, my name is David Elkins. I have a question for quantum\n> mechanics buffs out there. I am trying to learn the mathamatics behind\n> QM and am (not surprisingly) having some difficulties. One of the\n> difficulties is my confusion over what are vectors and vector spaces\n> in QM. In classical physics a vector is defined as a line segment with\n> magnitude and direction. In QM they use the term vector alot, but its\n> definition is somewhat different.\n> 1. Do vectors in QM have directionality?\n> 2. Do they have magnitude?\n> 3. Can they be represented as a line segment?\n> 4. What is vectorlike about them?\n> 5. Or is attempting to understand QM vectors classically a hopeless\n> endeavor?\n>\n\nIt isn\'t unusual to be confused about QM "vectors" the first time\nyou encounter them. I deal with students having this problem\nevery year. You are dealing with an abstraction of the classical\nvector concept, extending the properties of ordinary vectors to\nobjects with entirely different physical meanings. This\nextension takes some getting used to. Get a feel for what is\ngoing on first, and then learn the extra wrinkles and the formal\nmathematics.\n\nLet me use the wave function of a spinless particle in one\ndimension as an example of the use of vectors in QM. The\nproperties of ordinary vectors that are being generalized are\ntheir representation in terms of components, the fact that they\nadd, and their magnitude (question 2). An ordinary space vector\nhas 3 components, its x, y, and z projections. To specify the\nentire vector you must specify a number for each of the\ncomponents. A QM wave function phi(x) is specified by stating\nits value for all values of x. In thinking of the function as a\nvector, its value at each x is a component of the vector. The x\nlabels which component you are considering, and there are an\ninfinite number of components. Addition of two wave functions\nworks properly, since (phi1+phi2)(x) = phi1(x)+phi2(x), which in\nvector language states that components of the sum of two vectors\nis the sum of the components of the vector. The scalar product\nof two spatial vectors x and y is sum(xi X yi), so the scalar\nproduct of two wave function vectors is the sum over the\ncomponents of the product of the vectors. There is a\ncomplication here since wave functions are complex numbers and\nyou want the scalar product of a vector with itself to be real,\nand also because the sum over an infinite number of components in\nan integral. The proper generalization is that\nphi1.phi2 = integral [phi1*(x) phi2*(x) dx]. Now the magnitude\nof the wave-function vector should be integral [ |phi(x)|^2 dx ,\nthe normalization integral. Ordinarily, QM wave functions are\nall _unit vectors_.\n\nThe big question is, "why bother with the vector language?" The\nanswer is that the notation allows you to remember easily the\nformulas for changing the basis you are using for expansions of\nthe wave function and to recall quickly useful theorems about\nscalar products. At this point you might have to trust me on\nthat one, but the applications will show up soon enough.\n\nNow, your questions:\n1. Not necessarily\n2. Yes, and the magnitude represents the total probability of\nfinding the particle in the state represented by the wave function.\n3. They are mathematically analogous to a line segment, but the\nphysical analogy is almost never useful.\n4. Components and addition [and the other formal properties of\nvectors in the mathematical sense].\n5. Not hopeless at all, and potentially useful. Just make sure\nyou concentrate on analogies between the mathematical properties\nand not analogies between the physical pictures.\n\n> Lastly, on the side, can anyone give as concrete as possible\n> description of bra-ket notation? That\'s confusing the heck out\nof me.\n>\n\nA particle in a potential has many possible wave functions, which\ncan be labelled with constants called quantum numbers. When you\nlabel a wave-function with its quantum numbers, as for example\nphi/E\\ (x), the expression is tedious to write, especially when\nthere are a large number of labels. A good example of the latter\nis a hydrogen-atom wave function labelled with n, l, and m,\nits principal quantum numbers, the magnitude of its angular\nmomentum, and the z-projection of its angular momentum. |E> or\n|nlm> is easier to write than phi/E\\(x) or phi/nlm\\(r) and\ncontains the same information. As you use different expansions\nof the wave functions in terms of complete sets of functions\n(Fourier expansions) the bra-ket notation becomes much easier to\nuse than any notation using functions, and it will emphasize the\nuseful vector properties as well. To get used to the notation,\nmentally translate it to phi/E\\(x) for a while until you see the\nmeaning automatically. Then drop the use of functions for\nquicker understanding.\n> Thankx, David\n\nFeel free to ask more questions, either on or off the newsgroup.\n--\nMaurice Barnhill\nmvb@udel.edu [Use ReplyTo, not From]\n[bellatlantic.net is reserved for spam only]\nDepartment of Physics and Astronomy\nUniversity of Delaware\nNewark, DE 19716\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>davidelkins wrote:
> Hello, my name is David Elkins. I have a question for quantum
> mechanics buffs out there. I am trying to learn the mathamatics behind
> QM and am (not surprisingly) having some difficulties. One of the
> difficulties is my confusion over what are vectors and vector spaces
> in QM. In classical physics a vector is defined as a line segment with
> magnitude and direction. In QM they use the term vector alot, but its
> definition is somewhat different.
> 1. Do vectors in QM have directionality?
> 2. Do they have magnitude?
> 3. Can they be represented as a line segment?
> 4. What is vectorlike about them?
> 5. Or is attempting to understand QM vectors classically a hopeless
> endeavor?
>

It isn't unusual to be confused about QM "vectors" the first time
you encounter them. I deal with students having this problem
every year. You are dealing with an abstraction of the classical
vector concept, extending the properties of ordinary vectors to
objects with entirely different physical meanings. This
extension takes some getting used to. Get a feel for what is
going on first, and then learn the extra wrinkles and the formal
mathematics.

Let me use the wave function of a spinless particle in one
dimension as an example of the use of vectors in QM. The
properties of ordinary vectors that are being generalized are
their representation in terms of components, the fact that they
add, and their magnitude (question 2). An ordinary space vector
has 3 components, its x, y, and z projections. To specify the
entire vector you must specify a number for each of the
components. A QM wave function \phi(x) is specified by stating
its value for all values of x. In thinking of the function as a
vector, its value at each x is a component of the vector. The x
labels which component you are considering, and there are an
infinite number of components. Addition of two wave functions
works properly, since (phi1+phi2)(x) = phi1(x)+phi2(x), which in
vector language states that components of the sum of two vectors
is the sum of the components of the vector. The scalar product
of two spatial vectors x and y is sum(\xi X yi), so the scalar
product of two wave function vectors is the sum over the
components of the product of the vectors. There is a
complication here since wave functions are complex numbers and
you want the scalar product of a vector with itself to be real,
and also because the sum over an infinite number of components in
an integral. The proper generalization is that
phi1.phi2 = integral [phi1*(x) phi2*(x) dx]. Now the magnitude
of the wave-function vector should be integral [ |\phi(x)|^2 dx ,
the normalization integral. Ordinarily, QM wave functions are
all _unit vectors_.

The big question is, "why bother with the vector language?" The
answer is that the notation allows you to remember easily the
formulas for changing the basis you are using for expansions of
the wave function and to recall quickly useful theorems about
scalar products. At this point you might have to trust me on
that one, but the applications will show up soon enough.

Now, your questions:
1. Not necessarily
2. Yes, and the magnitude represents the total probability of
finding the particle in the state represented by the wave function.
3. They are mathematically analogous to a line segment, but the
physical analogy is almost never useful.
4. Components and addition [and the other formal properties of
vectors in the mathematical sense].
5. Not hopeless at all, and potentially useful. Just make sure
you concentrate on analogies between the mathematical properties
and not analogies between the physical pictures.

> Lastly, on the side, can anyone give as concrete as possible
> description of bra-ket notation? That's confusing the heck out
of me.
>

A particle in a potential has many possible wave functions, which
can be labelled with constants called quantum numbers. When you
label a wave-function with its quantum numbers, as for example
\phi/E\ (x), the expression is tedious to write, especially when
there are a large number of labels. A good example of the latter
is a hydrogen-atom wave function labelled with n, l, and m,
its principal quantum numbers, the magnitude of its angular
momentum, and the z-projection of its angular momentum. |E> or
|nlm> is easier to write than \phi/E\(x) or \phi/nlm\(r) and
contains the same information. As you use different expansions
of the wave functions in terms of complete sets of functions
(Fourier expansions) the bra-ket notation becomes much easier to
use than any notation using functions, and it will emphasize the
useful vector properties as well. To get used to the notation,
mentally translate it to \phi/E\(x) for a while until you see the
meaning automatically. Then drop the use of functions for
quicker understanding.
> Thankx, David

Feel free to ask more questions, either on or off the newsgroup.
--
Maurice Barnhill
mvb@udel.edu [Use ReplyTo, not From]
[bellatlantic.net is reserved for spam only]
Department of Physics and Astronomy
University of Delaware
Newark, DE 19716

Yi-Zen Chu; Yiren Qu

Aug13-04, 05:41 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\ndavidelkins wrote:\n> Hello, my name is David Elkins. I have a question for quantum\n> mechanics buffs out there. I am trying to learn the mathamatics behind\n> QM and am (not surprisingly) having some difficulties. One of the\n> difficulties is my confusion over what are vectors and vector spaces\n> in QM. In classical physics a vector is defined as a line segment with\n> magnitude and direction. In QM they use the term vector alot, but its\n> definition is somewhat different.\n\nThe definition of a vector space is a relatively intuitive axiomization\nof certain aspects of vectors in 3-D space that you\'ve mentioned above.\nI highly recommend that you simply pick up a linear algebra book written\nby mathematicians to learn from, if you have the time. The structure of\nvector spaces and the linear operators on them are usually not dealt in\na very rigorous and in-depth manner in physics. One fairly accessible\nbook I\'ve used when I was an undergraduate taking linear algebra is:\nFriedberg, Insel and Spence, Linear Algebra, Prentice Hall.\n\n> 1. Do vectors in QM have directionality?\n> 2. Do they have magnitude?\n> 3. Can they be represented as a line segment?\n> 4. What is vectorlike about them?\n> 5. Or is attempting to understand QM vectors classically a hopeless\n> endeavor?\n\nThe definition of a vector space itself does not contain ideas of length\nor angle. However this can be introduced for certain types of vector\nspaces (length and angle is always introduced in vector spaces in\nphysics), and in such a case the vector space becomes what is known in\nmathematical jargon as an inner product space. The bra-ket notation is\nDirac\'s method of introducing the inner product space into physics.\n\n> Lastly, on the side, can anyone give as concrete as possible\n> description of bra-ket notation? That\'s confusing the heck out of me.\n\nAlthough bra-ket notation is practically more useful - for instance\ncompleteness is simply expressed as Sum[n]|n><n| = 1, where {|n>} is a\nset of orthonormal basis vectors - I find the (a,b) notation used in\nmathematics to represent the inner product of vectors a and b to be\nclearer. Perhaps it would also help you to read the chapter on inner\nproduct space in the abovementioned book: once you are clear about the\ninner product as used in mathematics it\'d be easier to get used to the\nbra-ket notation because you\'d have a firm foundation to fall back on.\nThe only caution I\'d give is that in mathematics the inner product (,)\nis linear in the first component (ca,b)=c(a,b) and conjugate linear in\nthe second component (a,cb)=c*(a,b) whereas in physics the bra-ket is\nlinear in the ket-component and conjugate linear in the bra-component.\n\nYi-Zen\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>davidelkins wrote:
> Hello, my name is David Elkins. I have a question for quantum
> mechanics buffs out there. I am trying to learn the mathamatics behind
> QM and am (not surprisingly) having some difficulties. One of the
> difficulties is my confusion over what are vectors and vector spaces
> in QM. In classical physics a vector is defined as a line segment with
> magnitude and direction. In QM they use the term vector alot, but its
> definition is somewhat different.

The definition of a vector space is a relatively intuitive axiomization
of certain aspects of vectors in 3-D space that you've mentioned above.
I highly recommend that you simply pick up a linear algebra book written
by mathematicians to learn from, if you have the time. The structure of
vector spaces and the linear operators on them are usually not dealt in
a very rigorous and in-depth manner in physics. One fairly accessible
book I've used when I was an undergraduate taking linear algebra is:
Friedberg, Insel and Spence, Linear Algebra, Prentice Hall.

> 1. Do vectors in QM have directionality?
> 2. Do they have magnitude?
> 3. Can they be represented as a line segment?
> 4. What is vectorlike about them?
> 5. Or is attempting to understand QM vectors classically a hopeless
> endeavor?

The definition of a vector space itself does not contain ideas of length
or angle. However this can be introduced for certain types of vector
spaces (length and angle is always introduced in vector spaces in
physics), and in such a case the vector space becomes what is known in
mathematical jargon as an inner product space. The bra-ket notation is
Dirac's method of introducing the inner product space into physics.

> Lastly, on the side, can anyone give as concrete as possible
> description of bra-ket notation? That's confusing the heck out of me.

Although bra-ket notation is practically more useful - for instance
completeness is simply expressed as Sum[n]|n><n| = 1, where {|n>} is a
set of orthonormal basis vectors - I find the (a,b) notation used in
mathematics to represent the inner product of vectors a and b to be
clearer. Perhaps it would also help you to read the chapter on inner
product space in the abovementioned book: once you are clear about the
inner product as used in mathematics it'd be easier to get used to the
bra-ket notation because you'd have a firm foundation to fall back on.
The only caution I'd give is that in mathematics the inner product (,)
is linear in the first component (ca,b)=c(a,b) and conjugate linear in
the second component (a,cb)=c*(a,b) whereas in physics the bra-ket is
linear in the ket-component and conjugate linear in the bra-component.

Yi-Zen

Arnold Neumaier

Aug13-04, 05:42 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\n\ndavidelkins wrote:\n\n> Lastly, on the side, can anyone give as concrete as possible\n> description of bra-ket notation? That\'s confusing the heck out of me.\n\nIn the language of linear algebra, kets |psi> are just column vectors psi\n(for systems with finitely many levels only; each component gives the\namplitude for the corresponding level), and the corresponding bras\n<psi| are their complex conjugated transposed vectors psi^*.\nThe inner product <phi|psi> is therefore\n<phi|psi> = phi^*psi = sum_k phi^*k psi_k.\nFor the basis bra <k|, the unit vector with a single entry 1 at\nposition k, we find as special case\n<k|psi> = psi_k.\nIn infinite dimensions, the sum becomes an integral, and we get\n<phi|psi> = integral dx phi(x)^* psi(x)\nand for the basis bra <x|, which is a delta distribution centered at x,\npsi(x) = <x|psi>.\n\nActually, in infinite dimensions, one needs functional analysis\nin place of linear algebra to get a concise definition; kets are smooth\nfunction from some nice function space, and bras are linear functionals\non the dual space. The dual space is larger and also contains\ndistributions. (For those who want to be fully rigorous: kets belong to\na so-called nuclear space H_inf; its closure H under the Euclidean norm\ngives the conventional Hilbert space, and together with the dual\nH_inf^* = H_-inf, these define a Gelfand triple or rigged Hilbert space,\ntwo names for the same concept).\n\nPhysicists are less picky, however, and allow kets also to be\ndistributions, so that every bra has a corresponding ket. Thus they\nuse the ket |x> although this is not a smooth function.\nThis allows them to write not only psi(x) = <x|psi>, but also\npsi(x)^* = <x|psi>^* = <psi|x>.\nThe price to be paid is that inner products are no longer well-defined\nin general; for example, <x|x> is infinite. They say, |x> is not\nnormalizable and mean that it is not in the Hilbert space of well-behaved\npure states.\n\nCaution: Physicists often use different bases which may cause confusing\nnotation. For example <p| is a momentum basis state, while <x| is a\nposition basis state. But while <x|y> = 0 if x and y are distinct (positions),\nand <p|q> = 0 if p and q are distinct (momenta), the product of a\nmomentum bra <p| and a position ket |x> (or vice versa) is never zero.\n(Exercise: Verify this by computing explicit formulas for <p|x> and <x|p>!)\n\n\nThis explanation is now part of my theoretical physics FAQ at\nhttp://www.mat.univie.ac.at/~neum/physics-faq.txt\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>davidelkins wrote:

> Lastly, on the side, can anyone give as concrete as possible
> description of bra-ket notation? That's confusing the heck out of me.

In the language of linear algebra, kets |\psi> are just column vectors \psi
(for systems with finitely many levels only; each component gives the
amplitude for the corresponding level), and the corresponding bras
<\psi| are their complex conjugated transposed vectors \psi^*.
The inner product <\phi|\psi> is therefore
<\phi|\psi> = \phi^*\psi = sum_k \phi^*k \psi_k.
For the basis bra <k|, the unit vector with a single entry 1 at
position k, we find as special case
<k|\psi> = \psi_k.
In infinite dimensions, the sum becomes an integral, and we get
<\phi|\psi> = integral dx \phi(x)^* \psi(x)
and for the basis bra <x|, which is a \delta distribution centered at x,
\psi(x) = <x|\psi>.

Actually, in infinite dimensions, one needs functional analysis
in place of linear algebra to get a concise definition; kets are smooth
function from some nice function space, and bras are linear functionals
on the dual space. The dual space is larger and also contains
distributions. (For those who want to be fully rigorous: kets belong to
a so-called nuclear space H_{inf}; its closure H under the Euclidean norm
gives the conventional Hilbert space, and together with the dual
H_{inf}^* = H_-inf, these define a Gelfand triple or rigged Hilbert space,
two names for the same concept).

Physicists are less picky, however, and allow kets also to be
distributions, so that every bra has a corresponding ket. Thus they
use the ket |x> although this is not a smooth function.
This allows them to write not only \psi(x) = <x|\psi>, but also
\psi(x)^* = <x|\psi>^* = <\psi|x>.
The price to be paid is that inner products are no longer well-defined
in general; for example, <x|x> is infinite. They say, |x> is not
normalizable and mean that it is not in the Hilbert space of well-behaved
pure states.

Caution: Physicists often use different bases which may cause confusing
notation. For example <p| is a momentum basis state, while <x| is a
position basis state. But while <x|y> = if x and y are distinct (positions),
and <p|q> = if p and q are distinct (momenta), the product of a
momentum bra <p| and a position ket |x> (or vice versa) is never zero.
(Exercise: Verify this by computing explicit formulas for <p|x> and <x|p>!)

This explanation is now part of my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt

Arnold Neumaier

Igor Khavkine

Aug14-04, 10:26 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\n\njanelk@charter.net (davidelkins) wrote in message news:<294599a7.0408091940.5d4d62cd@posting.google. com>...\n> Hello, my name is David Elkins. I have a question for quantum\n> mechanics buffs out there. I am trying to learn the mathamatics behind\n> QM and am (not surprisingly) having some difficulties. One of the\n> difficulties is my confusion over what are vectors and vector spaces\n> in QM. In classical physics a vector is defined as a line segment with\n> magnitude and direction. In QM they use the term vector alot, but its\n> definition is somewhat different.\n\n> 4. What is vectorlike about them?\n\nQuantum mechanics uses abstract vectors. The only thing that\nis vector-like about that is the fact that you can add them\nand multiply them by scalars. Physically, saying that states\nare represented by vectors is equivalent to stating the superposition\nprinciple.\n\nI highly recommend picking up a basic linear algebra textbook\nand looking at the chapter on abstract vector spaces.\n\n> 1. Do vectors in QM have directionality?\n> 3. Can they be represented as a line segment?\n\nThese ideas have to do with pictorial representation of vectors,\nwhich works quite well for finite dimensional cases (finite number\nof components to a vector). However, it\'s a bit hard to apply it\nto vectors that must be described by an infinite number of components.\n\n> 2. Do they have magnitude?\n\nI suggest looking up "inner product space". The short answer is yes,\nand the magnitude is a straightforward generalization of the\nEuclidean vector length.\n\n> 5. Or is attempting to understand QM vectors classically a hopeless\n> endeavor?\n\nYou need neither classical nor quantum mechanics to understand vectors.\nJust like you need neither to understand addition and multiplication.\n\n> Lastly, on the side, can anyone give as concrete as possible\n> description of bra-ket notation? That\'s confusing the heck out of me.\n\nKets |n> are colmn vectors. Bras <n| are row vectors. You can go from\none to the other using transposition and conjugation. <n|m> is a\nnumber obtained by straight matrix multiplication. |n><m| is a matrix\nobtained by straight matrix multiplication. It is rather out of vogue\nin modern mathematics to think of linear algebra in terms of matrices.\nHowever, if you are already familiar with matrices, row and column\nvectors, they are a great way of keeping track of things. Just don\'t\nforget that there are many subtleties when vectors become infinite\ndimensional.\n\nHope this helps.\n\nIgor\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>janelk@charter.net (davidelkins) wrote in message news:<294599a7.0408091940.5d4d62cd@posting.google.com>...
> Hello, my name is David Elkins. I have a question for quantum
> mechanics buffs out there. I am trying to learn the mathamatics behind
> QM and am (not surprisingly) having some difficulties. One of the
> difficulties is my confusion over what are vectors and vector spaces
> in QM. In classical physics a vector is defined as a line segment with
> magnitude and direction. In QM they use the term vector alot, but its
> definition is somewhat different.

> 4. What is vectorlike about them?

Quantum mechanics uses abstract vectors. The only thing that
is vector-like about that is the fact that you can add them
and multiply them by scalars. Physically, saying that states
are represented by vectors is equivalent to stating the superposition
principle.

I highly recommend picking up a basic linear algebra textbook
and looking at the chapter on abstract vector spaces.

> 1. Do vectors in QM have directionality?
> 3. Can they be represented as a line segment?

These ideas have to do with pictorial representation of vectors,
which works quite well for finite dimensional cases (finite number
of components to a vector). However, it's a bit hard to apply it
to vectors that must be described by an infinite number of components.

> 2. Do they have magnitude?

I suggest looking up "inner product space". The short answer is yes,
and the magnitude is a straightforward generalization of the
Euclidean vector length.

> 5. Or is attempting to understand QM vectors classically a hopeless
> endeavor?

You need neither classical nor quantum mechanics to understand vectors.
Just like you need neither to understand addition and multiplication.

> Lastly, on the side, can anyone give as concrete as possible
> description of bra-ket notation? That's confusing the heck out of me.

Kets |n> are colmn vectors. Bras <n| are row vectors. You can go from
one to the other using transposition and conjugation. <n|m> is a
number obtained by straight matrix multiplication. |n><m| is a matrix
obtained by straight matrix multiplication. It is rather out of vogue
in modern mathematics to think of linear algebra in terms of matrices.
However, if you are already familiar with matrices, row and column
vectors, they are a great way of keeping track of things. Just don't
forget that there are many subtleties when vectors become infinite
dimensional.

Hope this helps.

Igor