Up and down arrows (↑ ↓) in LaTeX
\uparrow, \downarrow and \updownarrow draw vertical arrows (↑, ↓, ↕), used for monotonic convergence of sequences, spin states in physics, and vertical direction in diagrams.
LaTeX command
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In analysis, x_ is compact shorthand for 'the sequence x_n increases monotonically and converges to , with \downarrow used the same way for a decreasing sequence; this notation appears often in measure theory, for instance when stating monotone convergence results.
In quantum mechanics, \uparrow and \downarrow denote spin-up and spin-down states, typically written inside Dirac notation as \lvert \uparrow \rangle and \lvert \downarrow \rangle, a compact way to name the two basis states of a spin-1/2 system.
\updownarrow gives a single bidirectional vertical arrow, less common than the two single-direction arrows individually, used in diagrams to show something that can vary or move in either vertical direction. Double-line variants \Uparrow, \Downarrow and \Updownarrow also exist, in the same style as \Rightarrow.
Other forms
\downarrowdown (↓)\updownarrowup-down (↕)Examples
$x_n \uparrow x$a sequence increasing monotonically to its limit
$x_n \downarrow x$a sequence decreasing monotonically to its limit
$\lvert \uparrow \rangle$a spin-up state in Dirac notation
$a \updownarrow b$a bidirectional vertical relation in a diagram
Tips and common mistakes
- x_ is standard shorthand in real analysis for 'increases monotonically to', saving you from writing the full sentence out each time; pair it with
\downarrowfor the decreasing case. - For spin states, combine the arrow with Dirac notation,
\lvert \uparrow \rangleand\lvert \downarrow \rangle, rather than writing the arrow on its own. \Uparrow,\Downarrowand\Updownarroware the double-line versions, available with no extra package, useful in diagrams that already use\Rightarrow-style double arrows elsewhere.- Don't confuse
\uparrowwith\wedge(logical AND); despite a superficial resemblance in some fonts, they are unrelated commands with unrelated meanings.
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What does x_ mean?
That the sequence x_n increases monotonically and converges to x, a common shorthand in real analysis and measure theory.
How do I write spin-up and spin-down in LaTeX?
\uparrow and \downarrow, usually placed inside Dirac notation as \lvert \uparrow \rangle and \lvert \downarrow \rangle.
What's the difference between \uparrow and \Uparrow?
\uparrow is a single-line vertical arrow; \Uparrow is its double-line version, used where a bolder or implication-style vertical arrow is wanted.
What is \updownarrow used for?
A single bidirectional vertical arrow (↕), used in diagrams or notation where a quantity can move or point in either vertical direction.