Dirac notation and calculations

In summary, the conversation discusses a question about Dirac notation and the process of finding a normalised state. The participants discuss the concept of inner products and the calculation of bras and kets, ultimately determining that alpha should be equal to 1/sqrt(6) for the state vector to be normalised. They also mention the convention that alpha should be a real and positive number.
  • #1
electrogeek
12
1
Hello everyone,

I'm stuck on the question which I have provided below to do with Dirac notation:

In these questions |a>, |b> and |c> can be taken to form an orthonormal basis set
Consider the state |ξ> = α(|a> − 2|b> + |c>). What value of α makes |ξ> a normalised state?

I'm brand new to Dirac notation so I'm not too sure what to do? I was thinking that in order for |ξ> to be normalised, then (|ξ> )^2 = 1. But I don't know whether you can do (|ξ>)(|ξ>)? I've only ever seen something like <a|a> which equals 1 if you do <a|a> or 0 if you do <b|c>.

If you can square a ket, then would I be right in saying α is 1/ sqrt (3) because expressions like |a>|a> would be 1 and |a>|b> would be 0?

Any help will be greatly appreciated!
 
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  • #2
electrogeek said:
Hello everyone,

I'm stuck on the question which I have provided below to do with Dirac notation:

In these questions |a>, |b> and |c> can be taken to form an orthonormal basis set
Consider the state |ξ> = α(|a> − 2|b> + |c>). What value of α makes |ξ> a normalised state?

I'm brand new to Dirac notation so I'm not too sure what to do? I was thinking that in order for |ξ> to be normalised, then (|ξ> )^2 = 1. But I don't know whether you can do (|ξ>)(|ξ>)? I've only ever seen something like <a|a> which equals 1 if you do <a|a> or 0 if you do <b|c>.

Any help will be greatly appreciated!

Whether you're using Dirac notation of not, a state vector is normalised if its inner product with itself is 1. In Dirac notation this is

##\langle \xi|\xi \rangle = 1##
 
  • #3
PeroK said:
Whether you're using Dirac notation of not, a state vector is normalised if its inner product with itself is 1. In Dirac notation this is

##\langle \xi|\xi \rangle = 1##

Ah okay thank you. Are there any rules for calculating <ξ|? I know it's the hermitian conjugate but I don't really know how'd you calculate it for something like this. I know that if you treat kets like column vectors then bras are like row vectors bit that's it.
 
  • #4
electrogeek said:
Ah okay thank you. Are there any rules for calculating <a|? I know it's the hermitian conjugate but I don't really know how'd you calculate it for something like this. I know that if you treat kets like column vectors then bras are like row vectors bit that's it.

The basic rule for bras is that:

If ## | \xi \rangle = \alpha |a \rangle + \beta | b \rangle##

then ##\langle \xi |= \langle a| \alpha^* + \langle b| \beta^*##
 
  • #5
PeroK said:
The basic rule for bras is that:

If ## | \xi \rangle = \alpha |a \rangle + \beta | b \rangle##

then ##\langle \xi |= \langle a| \alpha^* + \langle b| \beta^*##

Ah okay. Thanks for the help. I'll give the question a go now.
 
  • #6
20191016_121705.jpg


I've just had a bit of free time to give the question a go and I got alpha to be 1/sqrt(6). I've attached my workings above - is this right?
 
  • #7
electrogeek said:
I've just had a bit of free time to give the question a go and I got alpha to be 1/sqrt(6). I've attached my workings above - is this right?

Yes. Note that, in simple terms you have a vector expressed as ##\alpha (1, -2, 1)## in an orthonormal basis. The magnitude of this vector is ##\alpha \sqrt{6}##. Hence, ##\alpha = 1/\sqrt{6}## for normalisation.

It's good to go through all the steps in Dirac notation, but it's only the usual linear algebra in the end.
 
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  • #8
PeroK said:
Yes. Note that, in simple terms you have a vector expressed as ##\alpha (1, -2, 1)## in an orthonormal basis. The magnitude of this vector is ##\alpha \sqrt{6}##. Hence, ##\alpha = 1/\sqrt{6}## for normalisation.

It's good to go through all the steps in Dirac notation, but it's only the usual linear algebra in the end.
Ah brilliant! Thanks for the help. Hopefully this will put me in a good position for the rest of the questions.
 
  • #9
electrogeek said:
Ah brilliant! Thanks for the help. Hopefully this will put me in a good position for the rest of the questions.

Just a minor additional point. This all assumes ##\alpha## is real and positive, which is the convention for normalisation constants. But, of course, any complex number with modulus ##1/\sqrt{6}## would do.
 
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  • #10
electrogeek said:
Hello everyone,

I'm stuck on the question which I have provided below to do with Dirac notation:

In these questions |a>, |b> and |c> can be taken to form an orthonormal basis set
Consider the state |ξ> = α(|a> − 2|b> + |c>). What value of α makes |ξ> a normalised state?

I'm brand new to Dirac notation so I'm not too sure what to do? I was thinking that in order for |ξ> to be normalised, then (|ξ> )^2 = 1. But I don't know whether you can do (|ξ>)(|ξ>)? I've only ever seen something like <a|a> which equals 1 if you do <a|a> or 0 if you do <b|c>.

If you can square a ket, then would I be right in saying α is 1/ sqrt (3) because expressions like |a>|a> would be 1 and |a>|b> would be 0?

Any help will be greatly appreciated!
You have the right idea. If you think of ##| a\rangle## and ##|b\rangle## as vectors in Hilbert space, then ##\langle a | b\rangle## is their dot product. So multiply out ##\langle \xi | \xi \rangle## and see what you get.
 

1. What is Dirac notation and how is it used in calculations?

Dirac notation, also known as bra-ket notation, is a mathematical notation used in quantum mechanics to represent quantum states and operators. It uses the symbols | and to represent a state vector and its dual vector, respectively. This notation is used to simplify calculations involving complex quantum systems.

2. How do you perform calculations using Dirac notation?

To perform calculations using Dirac notation, you first need to represent the quantum system in terms of state vectors and operators. Then, you can use the rules of Dirac notation to manipulate and combine these vectors and operators to obtain the desired result. This notation allows for efficient and elegant calculations in quantum mechanics.

3. What are the key properties of Dirac notation?

There are several key properties of Dirac notation that make it useful in quantum mechanics. These include linearity, orthogonality, and completeness. Linearity means that the notation follows the rules of linear algebra, allowing for simple manipulation of vectors and operators. Orthogonality means that vectors representing different states are perpendicular to each other, and completeness means that all possible states of a system can be represented using this notation.

4. How does Dirac notation simplify calculations in quantum mechanics?

Dirac notation simplifies calculations in quantum mechanics by providing a concise and intuitive way to represent quantum states and operators. It allows for easy manipulation and combination of vectors and operators, reducing the need for complex mathematical equations. This notation also allows for the use of linear algebra techniques, making calculations more efficient and elegant.

5. Are there any limitations to using Dirac notation in calculations?

While Dirac notation is a powerful tool in quantum mechanics, it does have some limitations. It is primarily used for non-relativistic quantum systems, as it does not take into account the effects of special relativity. Additionally, it may not be suitable for systems with a large number of particles, as the notation can become cumbersome and difficult to work with. In these cases, other mathematical techniques may be more appropriate.

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