# Proof for Problem 1.1.1 of Shankars Prin of QM

• Mindstein
In summary, the conversation discusses proving that |-V> = -|V> using the instruction to begin with |V> + (-|V>) = 0|V> = |0>. The participants mention the need to doubt everything and justify each step, as well as the concept of 1V = V as an axiom in linear spaces.
Mindstein
I think I may be over-thinking this as I have had one formal course in QM and an independent study course in QM, but any help is MORE THAN GREATLY APPRECIATED!

## Homework Statement

Prove that |-V> = -|V>

## Homework Equations

He instructs us to begin with |V> + (-|V>) = 0|V> = |0>

## The Attempt at a Solution

Let -|V> = |W>

Then

|V> + |W> = 0|V> = |0>

which implies that

|W> = |-V> ?

I'm really confused.

Okay, first I think you mean problem 1.1.2, unless you have a different edition.
This is the kind of proof where you have to doubt everything. Act as if you are in a mathematical minefield. Only do the things you can completely justify.

I find myself wanting to say that 1V = V, but that actually isn't an axiom...

Assuming that for the moment,
V + (-1)V = 1V + (-1)V = (1 + -1)V = 0V = 0 (by part 1 of the same problem)
And thus (-1)V is the unique vector -V such that V + -V = 0

Looking up in other books, 1V = V is listed as an eighth axiom for linear spaces. Shankar described it by interpretation in a paragraph, but didn't actually state it overtly as an axiom.

## 1. What is Problem 1.1.1 of Shankar's Principles of Quantum Mechanics?

Problem 1.1.1 of Shankar's Principles of Quantum Mechanics is a mathematical exercise that introduces the concept of the uncertainty principle in quantum mechanics. It involves calculating the uncertainties in position and momentum for a particle in a one-dimensional box.

## 2. Why is Problem 1.1.1 important in learning quantum mechanics?

Problem 1.1.1 is important because it helps students understand the fundamental principles of quantum mechanics, such as the uncertainty principle, which is a key concept in understanding the behavior of particles at the quantum level.

## 3. What is the solution to Problem 1.1.1?

The solution to Problem 1.1.1 involves using the Heisenberg uncertainty principle to calculate the uncertainties in position and momentum for a particle in a one-dimensional box. It is a mathematical solution that demonstrates the fundamental principles of quantum mechanics.

## 4. Are there any common mistakes that students make when solving Problem 1.1.1?

Yes, students often make mistakes when applying the mathematical principles to calculate the uncertainties in position and momentum. They may also struggle with understanding the conceptual implications of the uncertainty principle in quantum mechanics.

## 5. How can I prepare for solving Problem 1.1.1?

To prepare for solving Problem 1.1.1, it is important to have a strong understanding of the mathematical principles involved, such as the Heisenberg uncertainty principle. It is also helpful to review the conceptual implications of the uncertainty principle in quantum mechanics.

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